与3×3矩阵谱问题相联系的弧子方程族的拟周期解
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摘要
本文主要分为如下两个部分：其一,借助于Lenard递推序列及零曲率方程,推导出与偶的3×3超矩阵谱问题相联系的新的超KdV方程族和超KN方程族,并建立了它们的广义双Hamilton结构和无穷守恒律；其二,基于三角曲线理论及代数几何知识,构造了三族与3×3矩阵谱问题相联系的孤子方程的拟周期解.
第二章中,依据超李代数,将经典的2×2矩阵谱问题扩充为偶的3×3超矩阵谱问题.利用相容性条件,即零曲率方程,导出了新的超KdV方程族及超KN方程族.应用超迹公式讨论了这两族超孤子方程的Hamilton结构,并建立了超KdV方程及超KN方程的无穷守恒律.
我们知道,孤子方程的拟周期解不仅揭示了解的内部结构,描述了非线性现象的拟周期行为,或孤子方程的可积性特征,而且可以利用它约化出多孤子解,椭圆函数解及其它形式的解.因此,研究孤子方程的拟周期解就变得十分重要.本文第三章到第五章,分别讨论了三族与3×3矩阵谱问题相联系的孤子方程的拟周期解.通过方程族的Lax矩阵的特征多项式,定义了一条三角曲线κg,并将其紧化为亏格为9的三叶Riemann面.在此Riemann面上引入适当的Baker-Akhiezer函数,亚纯函数及椭圆变量,从而将孤子方程分解为可解的Dubrovin-type常微分方程系统.然后,在Abel映射下流被拉直.进一步,根据亚纯函数及B aker-Akhiezer函数零点和奇点的性质,利用第二类和第三类Abel微分.Riemann定理及Riemann-Roch定理得到了它们的Riemann theta函数表示.最后,结合亚纯函数或(及)Baker-Akhiezer函数的Riemann theta函数表示和它们的渐近性质,便给出了孤子方程族的拟周期解.
The thesis can be mainly divided into two parts. First, with the help of Lenard recursion equations and the zero-curvature equation, we derive a new super KdV hierarchy and super KN hierarchy related to two even3×3matrix spectral problems. Moreover, generalized bi-Hamiltonian structures and infinite conservation laws of these two hierarchies are established; On the other hand, based on the theory of trigonal curve and the knowledge of algebraic geometry, we construct the quasi-periodic solutions of three hierarchies of soliton equations associated with three3×3matrix spectral problems.
In chapter two, by extending the corresponding classical2×2spectral problems to the even3×3super matrix spectral problems according to super Lie algebra, we propose a new super KdV hierarchy and super KN hierarchy resorting to the compatible condition i.e. zero-curvature equation. Applying the super trace iden-tity, we discuss the generalized bi-Hamlitonian structures of the two super soliton hierarchies. Furthermore, we establish the infinite sequence of conserved quantities of the super KdV equation and super KN equation.
As we all know, quasi-periodic solutions of soliton equations not only reveal in-herent structure mechanism of solutions and describe the quasi-periodic behavior of nonlinear phenomenon or characteristic for the integrability of soliton equations, but also can be reduced to find multi-soliton solutions, elliptic function solutions, and others. Therefore, the research on the quasi-periodic solutions of soliton equations is of greatest importance. From chapter three to five, we discuss the quasi-periodic solutions of three hierarchies of soliton equations associated with three different3×3matrix spectral problems, respectively. With the aid of the characteristic polynomial of Lax matrix for soliton hiearachy, we define a trigonal curve κg, and then its com- pactification becomes a three-sheeted Riemann surface of arithmetic genus g. We introduce the appropriate Baker-Akhiezer function, meromorphic function and ellip-tic variables on the three-sheeted Riemann surface, from which soliton equations are decomposed into the system of solvable Dubrovin-type ordinary differential equa-tions. Then, under the Abel map, the flows of soliton hierarchy are straightened. Furthermore, in accordance with the properties of the zeros and singularities of the meromorphic function and Baker-Akhiezer function, we get their Riemann theta function representations by means of the second and third Abel differentials, Rie-mann theorem and Riemann-Roch theorem. Combing the Riemann theta function representations of the meromorphic function or(and) the Baker-Akhiezer function with their asymptotic properties, we finally obtain the quasi-periodic solutions of soliton equations.
第二章中,依据超李代数,将经典的2×2矩阵谱问题扩充为偶的3×3超矩阵谱问题.利用相容性条件,即零曲率方程,导出了新的超KdV方程族及超KN方程族.应用超迹公式讨论了这两族超孤子方程的Hamilton结构,并建立了超KdV方程及超KN方程的无穷守恒律.
我们知道,孤子方程的拟周期解不仅揭示了解的内部结构,描述了非线性现象的拟周期行为,或孤子方程的可积性特征,而且可以利用它约化出多孤子解,椭圆函数解及其它形式的解.因此,研究孤子方程的拟周期解就变得十分重要.本文第三章到第五章,分别讨论了三族与3×3矩阵谱问题相联系的孤子方程的拟周期解.通过方程族的Lax矩阵的特征多项式,定义了一条三角曲线κg,并将其紧化为亏格为9的三叶Riemann面.在此Riemann面上引入适当的Baker-Akhiezer函数,亚纯函数及椭圆变量,从而将孤子方程分解为可解的Dubrovin-type常微分方程系统.然后,在Abel映射下流被拉直.进一步,根据亚纯函数及B aker-Akhiezer函数零点和奇点的性质,利用第二类和第三类Abel微分.Riemann定理及Riemann-Roch定理得到了它们的Riemann theta函数表示.最后,结合亚纯函数或(及)Baker-Akhiezer函数的Riemann theta函数表示和它们的渐近性质,便给出了孤子方程族的拟周期解.
The thesis can be mainly divided into two parts. First, with the help of Lenard recursion equations and the zero-curvature equation, we derive a new super KdV hierarchy and super KN hierarchy related to two even3×3matrix spectral problems. Moreover, generalized bi-Hamiltonian structures and infinite conservation laws of these two hierarchies are established; On the other hand, based on the theory of trigonal curve and the knowledge of algebraic geometry, we construct the quasi-periodic solutions of three hierarchies of soliton equations associated with three3×3matrix spectral problems.
In chapter two, by extending the corresponding classical2×2spectral problems to the even3×3super matrix spectral problems according to super Lie algebra, we propose a new super KdV hierarchy and super KN hierarchy resorting to the compatible condition i.e. zero-curvature equation. Applying the super trace iden-tity, we discuss the generalized bi-Hamlitonian structures of the two super soliton hierarchies. Furthermore, we establish the infinite sequence of conserved quantities of the super KdV equation and super KN equation.
As we all know, quasi-periodic solutions of soliton equations not only reveal in-herent structure mechanism of solutions and describe the quasi-periodic behavior of nonlinear phenomenon or characteristic for the integrability of soliton equations, but also can be reduced to find multi-soliton solutions, elliptic function solutions, and others. Therefore, the research on the quasi-periodic solutions of soliton equations is of greatest importance. From chapter three to five, we discuss the quasi-periodic solutions of three hierarchies of soliton equations associated with three different3×3matrix spectral problems, respectively. With the aid of the characteristic polynomial of Lax matrix for soliton hiearachy, we define a trigonal curve κg, and then its com- pactification becomes a three-sheeted Riemann surface of arithmetic genus g. We introduce the appropriate Baker-Akhiezer function, meromorphic function and ellip-tic variables on the three-sheeted Riemann surface, from which soliton equations are decomposed into the system of solvable Dubrovin-type ordinary differential equa-tions. Then, under the Abel map, the flows of soliton hierarchy are straightened. Furthermore, in accordance with the properties of the zeros and singularities of the meromorphic function and Baker-Akhiezer function, we get their Riemann theta function representations by means of the second and third Abel differentials, Rie-mann theorem and Riemann-Roch theorem. Combing the Riemann theta function representations of the meromorphic function or(and) the Baker-Akhiezer function with their asymptotic properties, we finally obtain the quasi-periodic solutions of soliton equations.
引文
[1]J. S. Russell, Report of the committee on waves, Report of the 7th Meeting of British Association for the Advancement of Science, Liverpool (1838) 417-496.
[2]J. S. Russell, Report on waves, Report of the 14th Meeting of British Association for the Advancement of Science, John Murray, London (1844) 311-390.
[3]J. Boussinesq, Theorie de lintumescence liquid appelee onde solitaire ou de teansla-tion, se propagente dans un canal retangulaire, Compte Rendus Acad. Sci. Paris,72 (1871) 755-759.
[4]J. Boussinesq, Theorie des ondes et de remous qui se propagent le long dun canal rect-angulaire horizontal, en communiquant au liquide contenu dans ce canal des vitesses sensiblemant parielles de la surface au fond, J. Math. Pure Appl. Ser.2,17 (1872) 55-108.
[5]D. J. Korteweg and G. de Vries, On the change of form of long waves advancing in a rectangular canal, an on a new type of long stationary wave, Philos. Mag. Ser.39 (1895) 422-443.
[6]E. Fermi, J. Pasta and S.Ulam, Studies of nonlinear problems, Los Alamos REP. LA (1955) 143-156.
[7]M. Toda, Vibration of chain with nonlinear interaction, J. Phys. Soc. Jpn.22 (1967) 431-436.
[8]N. J. Zabusky and M. D. Kruskal, Interaction of solitons in a collisionless plasma and the recurrence of initial status, Phys. Rev. Lett.15 (1965) 240-243.
[9]C. S. Gardner, J. M. Greene, M. D. Kruskal and R. M. Miura, Method for solving the Korteweg-de Vries equation, Phys. Rev. Lett.19 (1967) 1095-1097.
[10]C. S. Gardner, J. M. Greene, M. D. Kruskal and R. M. Miura, The Korteweg de Vries equation and generalizations. VI. Methods for exact solution, Comm. Pure Appl. Math.21 (1974) 97-133.
[11]M. J. Ablowitz, D.J. Kaup, A.C. Newell and H. Segur, The inverse scattering transform-Fourier analysis foe nonlinear problems, Stud. Appl. Math.53 (1974) 249-315.
[12]M. J. Ablowitz and H. Segur, Solitons and the inverse scattering transform (PA: SIAM, Philadelphia,1981).
[13]M. J. Ablowitz and P. A. Clarkson, Solitons, nonlinear evolution equations and in-verse scattering, London Mathematical Society Lecture Notes Series 149, (Cambridge University Press 2000).
[14]S. Novikov, S. V. Manakov, L. P. Pitaevskii and V. E. Zakharov, Theory ofsolitons, the inverse scattering methods (Consultants Bureau, New York,1984).
[15]R. Beals and R. R. Coifman, Inverse scattering and evolution equations, Commun. Pure Appl. Math.38 (1985) 29-42.
[16]H. Flaschka, On the Toda lattice II. Inverse scattering solutions, Prog. Theor. Phys. 51 (1974) 703-716.
[17]S. V. Manakov, Complete integrability and stochastization of discrete dynamical sys-tems, Sov. Phys. JETP Lett.40 (1975) 269-274.
[18]B. B. Kadomtsev and V. I. Petviashvili, On the stability of solitary waves in weakly dispersive media, Sov. Phys. Dokl.15 (1970) 539-541.
[19]A. Davey and K. Stewartson, On three-dimensional packets of surface waves, Proc. Roy. Soc. London Ser. A 338 (1974) 101-110.
[20]V. E. Zakharov and A. B. Shabat, A scheme for integrating the nonlinear equations of mathematical physics by the method of the inverse scattering problem. I, Funct. Anal. Appl.8 (1974) 226-235.
[21]V. E. Zakharov and A. B. Shabat, Integration of nonlinear equations of mathematical physics by the method of inverse scattering. Ⅱ, Funct. Anal. Appl.13 (1979) 166-174.
[22]P. D. Lax, Integrals of nonlinear equations of evolution and solitary waves, Commun. Pure Appl. Math.21 (1968) 467-490.
[23]R. Hirota, Exact solution of the sine-Gordon equation for mutiple collisions of solitons, J. Phys. Soc. Japan 33 (1972) 1459-1463.
[24]R. Hirota, Exact solution of the Korteweg-de Vries equation for mutiple collisions of solitons, Phys. Rev. Lett.27 (1971) 1192-1194.
[25]R. Hirota, The direct method in soliton theory (Cambridge University Press, Cam-bridge,2004).
[26]X. B. Hu, Nonlinear superposition formula of the Novikov-Veselov equation, J. Phys. A:Math. Gen.27 (1994) 1331-1338.
[27]X. B. Hu, Nonliear superposition formulate for the differential-difference analogue of the KdV equation and two-dimensional Toda equation, J. Phys. A:Math. Gen.27 (1994) 201-214.
[28]X. B. Hu, The higher-order KdV equation with a source and nonlinear superposition formula, Chaos Solitons and Fractals 7 (1996) 211-215.
[29]D. J. Zhang, The N-soliton solutions for the modified KdV equation with self-consistent sources, J. Phys. Soc. Jpn.71 (2002) 2649-2656.
[30]D. J. Zhang and D. Y. Chen, The N-soliton solutions of the sine-Gordon equation with self-consistent sources, Physica A 321 (2003) 467-481.
[31]J. Hietarinta and D. J. Zhang, Multisoliton solutions to the lattice Boussinesq equa-tion, J. Math. Phys.51 (2010) 033505.
[32]V. B. Matveev and M. A. Salle, Darboux transformations and solitons (Springer,
[33]谷超豪,胡和生,周子翔,孤立子理论中的达布变换及其几何应用(上海科学技术出版社,上海,1999).
[34]X. G. Geng and H. W. Tam, Darboux transformation and soliton solutions for gen-eralized nonlinear Schrodinger equations, J. Phys. Soc. Jpn.68 (1999) 1508-1512.
[35]E. G. Fan, Darboux transformation for soliton-like solutions for the Gerdjikov-Ivanov equatiion, J. Phys. A:Math. Gen.33 (2000) 6925-6933.
[36]Y. S. Li and J. E. Zhang, Darboux transformations of classical Boussonesq syetem and its new solutions, Phys. Lett. A.284 (2001) 253-258.
[37]C. Rogers and W. K. Schief, Backlund and Darboux transformations geometry and modern applcations in soliton theory, (Cambridge University Press, Cambridge, 2002).
[38]M. Bordemann, M. forger, J. Laartz and U. Schaper, The Lie-Poisson structure of integrable classical nonlinear sigma models, Comm. Math. Phys.152 (1993) 167-190.
[39]B. Jovanovic, Non-holonomic geodesic flows on Lie groups and the integrable Suslov problem on SO(4), J. Phys. A:Math. Gen.31 (1998) 1415-1422.
[40]R. Z. Zhdanov and V. I. Lahno, Group classification of heat conductivity equations with a nonlinear source, J. Phys. A:Math. Gen.32 (1999) 7405-7418.
[41]S. Spichak and V. Stognii, Symmetry classification and exact solutions of the one-dimensional Fokker-Planck equation with arbitrary coefficients of drift and diffusion, J. Phys. A:Math. Gen.32 (1999) 8341-8353.
[42]D. L. Du, Complex form, reduction and Lie-Poisson structure for the nonlinearized spectral problem of the Heisenberg hierarchy, Physica A 303 (2001) 439-456.
[43]C. W. Cao, Nonlinearization of the Lax system for AKNS hierarchy, Science China A 33 (1990) 528-536.
[44]C. W. Cao and X. G. Geng, C Neumann and Bargmann systems associated with the coupled KdV soliton hierarchy, J. Phys. A:Math. Gen.23 (1990) 4117-4125.
[45]C. W. Cao, A classical integrable system and the involutive representation of solutions of the KdV equation, Acta. Math. Sinica 7 (1991) 216-223.
[46]C. W. Cao and X. G. Geng, A nonconfocal generator of involutive systems and three associated soliton hierarchies, J. Math. Phys.32 (1991) 2323-2328.
[47]X. G. Geng, Discrete Bargmann and Neumann systems and finite-dimensional inte-grable systems, Physica A 212 (1994) 132-142.
[48]C. W. Cao, X. G. Geng and Y. T. Wu, From the special (2+1)-Toda lattice to the Kadomtsev-Petviashvili equation, J. Phys. A:Math. Gen.32 (1999) 8059-8078.
[49]X. G. Geng and H. H. Dai, Nonlinearization of the eigenvalue problem related to coupled nonlinear Schrodinger equations, J. Math. Anal. Appl.233 (1999) 26-55.
[50]X. G. Geng and C. W. Cao, Decomposition of the (2+1)-dimensional Gardner equa-tion and its quasi-periodic solutions, Nonlinearity 14 (2001) 1433-1452.
[51]R. G. Zhou, Nonlinearizations of spectral problems of the nonlinear Schrodinger equa-tion and the real-valued modified Korteweg-de Vries equation, J. Math. Phys.48 (2007) 013510.
[52]R. G. Zhou, Relationships among three kinds of finite dimensional integrable Hamil-tonian systems, Physica A 320 (2003) 128-140.
[53]C. W. Cao, Parametric representation of the finite-band solution of the Heisenberg equation, Phys. Lett. A.184 (1994) 333-338.
[54]X. G. Geng, C. W. Cao and H. H. Dai, Quasi-periodic solutions for some (2+1)-dimensional integrable models generated by the Jaulent-Miodek hierarchy, J. Phys. A:Math. Gen.34 (2001) 989-1004.
[55]C. W. Cao, X. G. Geng and H. Y. Wang, Algebraic-geometric solution of the (2+1)-dimensional Burgers equatiion with a discrete variable, J. Math. Phys.43 (2002) 621-623.
[56]C. W. Cao, Y. T. Wu, and X. G. Geng, Relation between the Kadomtsev-Petviashvili equation and the Confocal involutive system, J. Math. Phys.40 (1999) 3948-3970.
[57]X. G. Geng, Y. T. Wu and C. W. Cao, Quasi-periodic solutions of the modified Kadomtsev-Petviashvili equation, J. Phys. A:Math. Gen.32 (1999) 3733-3742.
[58]X. G. Geng and Y. T. Wu, Finite-band solutions of the classical Boussinesq-Burgers equations, J. Math. Phys.40 (1999) 2971-2982.
[59]X. G. Geng, H. H. Dai and C. W. Cao, Algebro-geometric constructions of the discrete Ablowitz-Ladik flows and applications, J. Math. Phys.44 (2003) 4573-4588.
[60]X. G. Geng, Algebraic-geometrical solutions of some multidimensional nonlinear evo-lution equations, J. Phys. A:Math. Gen.36 (2003) 2289-2303.
[61]H. H. Dai and X. G. Geng, Explicit solutions of the (2+1)-dimensional modified Toda Lattice through straightening out of the relativistic Toda flows, J. Phys. Soc. Jpn.72 (2003) 3063-3069.
[62]X. G. Geng, H. H. Dai, and J. Y. Zhu, Decomposition of the discrete Ablowitz-Ladik hierarchy, Stud. Appl. Math.118 (2007) 281-312.
[63]Z. J. Qiao, Generalized r-matrix structure and algebro-geometric solution for inte-grable systems, Rev. Math. Phys.21 (2001) 545-586.
[64]Z. J. Qiao, The Camassa-Holm hierarchy, N-dimensional integrable systems, and algebro-geometric solution on a symplectic submanifold, Comm. Math. Phys.239 (2003) 309-341.
[65]V. E. Zakharov and S. V. Manakov, Generalization of the method of inverse scattering problem, Teoret. Mat. Fiz.27 (1976) 283-287.
[66]E. A. Kuznetsov, M. D. Spector and G. E. Falkovich, On the stability of nonlinear waves in integrable models, Physica D 10 (1984) 379-386.
[67]A. I. Zenchuk, A unified dressing method for C-integrable hierarchies; the particular example of a (3+1)-dimensional n-wave equation, J. Phys. A:Math. Gen.37 (2004) 6557-6571.
[68]H. H. Dai and A. Jeffery, The inverse scattering transforms for certain types of variable coefficient KdV equations, Phys. Lett. A 139 (1989) 369-372.
[69]A. Jeffery and H. H. Dai, On the application of a generalized version of the dressing method to the integration of variable coefficient KdV equation, Rendiconti di Math., Serie VII 10(1990) 439-455.
[70]J. Y. Zhu and X. G. Geng, The generalized version of dressing method with applica-tions to AKNS equations, J. Nonlinear Math. Phys.13 (2006) 81-89.
[71]M. J. Ablowitz, A. Ramani and H. Segur, A connection between nonlinear evolution equations and ordinary diefrential equations of p-type I, J. Math. Phys.21 (1980) 715-721.
[72]M. J. Ablowitz, A. Ramani and H. Segur, A connection between nonlinear evolution equations and ordinary diefrential equations of p-type II, J. Math. Phys.21 (1980) 1006-1015.
[73]J. B. McLeod and P. J. Olver, The connection between patial differential equations soluble by inverse scattering and odinary dieferntial equations of painleve type, J. Math. Anal.14 (1983) 488-506.
[74]R. Conte, Invariant painleve analysis of Partial difereniial equations, Phys. Lett. A 140 (1989) 383-390.
[75]S. Y. Lou and Q. X. Wu, Painleve integrability of two sets of nonlinear evolution equations with nonlinear dispersions, Phys. Lett. A 262 (1999) 344-349.
[76]S. L. Zhang, B. Wu and S. Y. Lou, Painleve analysis and special solutions of gener-alized Broer-Kaup equations, Phys. Lett. A.300 (2002) 40-48.
[77]Y. Cheng and Y. S. Li, The constraint of the Kadometsev-Petviashvili equation and its special solutions, Phys. Lett. A 157 (1991) 22-26.
[78]Y. Cheng and Y. S. Li, Constraints of the (2+l)-dimensional integrable soilton sys-tems, J. Phys. A:Math. Gen.25 (1992) 419-431.
[79]S. Y. Lou and J. Lin, The generalized symmetry algebraic of the bilinear Kadomtsev-Petviashvili equation, Phys. Lett. A 185 (1994) 29-34.
[80]S. Y. Lou and X. B. Hu, Infinitely many Lax pairs and symmetry constraints of the KP equation, J. Math. Phys.38 (1997) 6401-6427.
[81]C. Z. Qu, Symmetries and solutions to the thin film equations, J. Math. Anal. Appl. 317 (2006) 381-397.
[82]C. Z. Qu, S. L. Zhang and R. C. Liu, Separation of variables and exact solutions to quasilinear diffusion equations with nonlinear source, Physica D 144 (2000) 97-123.
[83]P. G. Estevez, C. Z. Qu and S. L. Zhang, Separation of variables of a generalized porous medium equation with nonlinear source, J. Math. Anal. Appl.275 (2002) 44-59.
[84]J. Y. Hu and C. Z. Qu, Functionally separable solutions to nonlinear wave equations by group foliation method, J. Math. Anal. Appl.330 (2007) 298-311.
[85]M. L. Wang, Exact solutions for a compound KdV-Burgers equation, Phys. Lett. A 213 (1996) 279-287.
[86]M. L. Wang and Y. M. Wang, A new Backlund transformation and multi-solutions to the KdV equation with general variable coefficients, Phys. Lett. A 287 (2001) 211-216.
[87]R. G. Zhou, The finite-band solutions of the Jaulent-Miodek equation, J. Math. Phys. 38 (1997) 2535-2546.
[88]R. G. Zhou, Integrable Rosochatius deformations of the restricted soliton flows, J. Math. Phys.48 (2007) 2535-2551.
[89]Z. J. Qiao, G. P. Zhang, on peaked and smooth solitons for the Camassa-Holm equa-tion, Europhys. Lett.73 (2006) 657-663.
[90]Z. J. Qiao, The matrix and an algebraic-geometric solution of the AKNS system, Theore. Math. Phys.127 (2001) 827-834.
[91]J. Wess and B. Zumino, Supergauge transformations in four dimensions, Nucl. Phys, B 70 (1974) 39-50.
[92]V. G. Kac, A Sketch of Lie Superalgebra Theory, Commun. Math. Phys.53 (1977) 31-64.
[93]V. G. Kac, Lie super algebras, Adv. Math.26 (1977) 8-96.
[94]F. A. Berezin and D. A. Leites, Supermanifolds, Dokl. Akad. Nauk SSSR 224 (1975) 505-508.
[95]V. Rittenberg, A guide to Lie superalgebras, Group Theoretical Methods in Physics 79 (1978) 3-21.
[96]D. A. Leites, Lie superalgebras, Funct. Anal.9 (1975) 75-76.
[97]D. A. Leites, Introduction to the theory of supermanifolds, Russ. Math. Surv.35 (1980) 1-64.
[98]F. Berezin, Introduction to super analysis, (D. Reidel Publishing Company, Dor-drecht,1987).
[99]B. S. Dewitt, Supermanifolds (Cambrige University Press,1992).
[100]F. A. Rogers, Supermanifolds:theorey and application, (World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ,2007).
[101]B. A. Kupershmidt, A super Korteweg-de Vries equation:an integrable system, Phys. Lett. A 102 (1984) 213-215.
[102]B. A. Kupershmidt, Bosons and fermions interacting integrably with the Korteweg-de Vries field, J. Phys. A:Math. Gen.17 (1984) L869-L872.
[103]B. A. Kupershmidt, Super Korteweg-de Vries equation associated to super extension of virasoro algebra, Phys. Lett. A 109 (1985) 417-423.
[104]M. Gurses and O. Oguz, A super AKNS scheme, Phys. Lett. A 108 (1985) 437-440.
[105]M. Gurses and O. Oguz, A super soliton connection, Lett. Math. Phys.11 (1986) 235-246.
[106]Y. S. Li and L. N. Zhang, Super AKNS scheme and its infinite conserved currents, Nuovo Cimento A 93 (1986) 175-183.
[107]Y. S. Li and L. N. Zhang, A note on the super AKNS equations, J. Phys. A:Math. Gen.21 (1988) 1549-1552.
[108]Y. S. Li and L. N. Zhang, Hamiltonian structure of the super evolution equation, J. Math. Phys.31 (1990) 470-475.
[109]X. B. Hu, An approach to generate superextentions of integrable systems, J. Phys. A:Math. Gen.30 (1997) 619-632.
[110]M. Antonowicz and A. P. Fordy, Super-Extensions of Energy Dependent Sehrodinger Operators, Commun. Math. Phys.124 (1989) 487-500.
[Ill]Yu. I. Manin and A.0. Radul, A supersymmetric extension of the Kadomtsev-Petviashvili hierarchy, Commun. Math. Phys.98 (1985) 65-77.
[112]P. Mathieu, Supersymmetric extension of the Korteweg-de Vries Equation, J. Math. Phys.29 (1988) 2499-2506.
[113]Z. Popowicz, The fully supersymmetric AKNS equations, J. Phys. A:Math. Gen. 23 (1990) 1127-1136.
[114]Z. Popowicz, A 2-component or N= 2 supersymmetric Camassa-Holm equation, Phys. Lett. A 354 (2006) 110-114.
[115]Z. Popowicz, Odd Hamiltonian structure for supersymmetric Sawada-Kotera equa-tion, Phys. Lett. A 373 (2009) 3315-3323.
[116]Q. P. Liu, Supersymmetric Harry-Dym type equations, J. Phys. A:Math. Gen.28 (1995) L245-L248.
[117]Q. P. Liu, Ziemowit Popowicz and K. Tian, The even and odd supersymmetric Hunter-Saxton and Liouville equations, Phys. Lett. A 375 (2010) 29-35.
[118]K. Tian and Q. P. Liu, A supersymmetric Sawada-Kotera equation, Phys. Lett. A 373 (2009) 1807-1810
[119]Q. P. Liu, Fully supersymmetric hierarchies from an energy-dependent super Hill operator, J. Phys. A:Math. Gen.30 (1997) 8661-8670.
[120]G. H. M. Roelofs and P. H. M. Kersten, Supersymmetric extensions of the nonlinear Sehrodinger equation:Symmetries and coverings, J. Math. Phys.33 (1992) 2185-2206.
[121]S. Q. LU, X. B. Hu and Q. P. Liu, A supersymmetric Ito's equation and its soliton solutions, J. Phys. Soc. Jpn.75 (2006) 064004.
[122]J. C. Beuneli, A. Das snd Z. Popowicz, Supersymmetric extensions of the Harry Dym hierarchy, J. Math. Phys.44 (2003) 4756-4767.
[123]C. M. Yung, The N=2 supersymmetric Boussinesq hierarchies, Phys. Lett. B 309 (1993) 75-84.
[124]J. Lenells and O. Lechtenfeld, On the N=2 supersymmetric Camassa-Holm and Hunter-Saxton equations, J. Math. Phys.50 (2009) 012704.
[125]M. A. Olshanetsky, Supersymmetric Two-Dimensional Toda Lattice, Commun. Math. Phys.88 (1983) 63-76.
[126]K. Ikeda, A Supersymmetric Extension of the Toda Lattice Hierarchy, Lett. Math. Phys.14 (1987) 321-328.
[127]P. Mathieu, Hamiltonian structure of graded and super evolution equations, Lett. Math. Phys.16 (1988) 199-206.
[128]Q. P. Liu, Bi-Hamiltonian structure of a genralized super Korteweg-de Vrie equation, J. Phys. A:Math. Gen.26 (1993) L1239-L1242.
[129]Y. Watanabe, A Hamilton Formalism for Evolution Equations with Odd Variables, Annali di Matematica pura ed applicata (IV), Vol. CLIV (1989) 127-146.
[130]W. Oevel and Z. Popowicz, The Bi-Hamiltonian Structure of Fully Supersymmetric Korteweg-de Vries Systems, Commun. Math. Phys.139 (1991) 441-460.
[131]J. F. Gomes, L. H. Ymai, A. H. Zimerman, Permutability of Backlund transforma-tion for N=1 supersymmetric sinh-Gordon, Phys. Lett. A 373 (2009) 1401-1404.
[132]A. R. Chowdhury and S. Roy, On the Backlund transformation and Hamiltonian properties of superevaluation equations, J. Math. Phys.27 (1986) 2464-2468.
[133]Q. P. Liu, Darboux transformations for supersymmetric Korteweg-de Vries equa-tions, Lett. Math. Phys.35 (1995) 115-122.
[134]Q. P. Liu and M. Manas, Darboux transformation for the Manin-Radul supersym-metric KdV equation, Phys. Lett. B 394 (1997) 337-342.
[135]Q. P. Liu and M. Manas, Darboux transformation for supersymmetric Kadomtsev-Petviashvili hierarchy, Phys. Lett. B 485 (2000) 293-300.
[136]M. Siddiq, M. Hassan and U. Saleem, On Darboux transformation of the supersym-metric sine-Gordon equation, J. Phys. A:Math.Gen.39 (2006) 7313-7318.
[137]A. S. Carstea, Extension of the bilinear formalism to supersymmetric KdV-type equations, Nonlinearity 13 (2000) 1645-1656.
[138]A. S. Carstea, Bilinear approach to supersymmetric KdV equation, J. Nonlinear Math. Phys.8 (Suppl.) (2001) 48-52.
[139]Q. P. Liu and X. B. Hu, Biliearization of N= 1 supersymmetric Korteweg-de Vries equation revisited, J. Phys. A:Math. Gen.38 (2005) 6371-6378.
[140]Q. P. Liu, X. B. Hu and M. X. Zhang, Supersymmetric modified Korteweg-de Vries equation:bilinear approach, Nonlinearity 18 (2005) 1597-1603.
[141]E. G. Fan, Supersymmetric KdV-Sawada-Kotera-Ramani equation and its quasi-periodic wave solutions, Phys. Lett. A 374 (2010) 744-749.
[142]E. G. Fan and Y. C. Hon, Quasiperiodic Wave Solutions of N=2 Supersymmetric KdV Equation in Superspace, Stud. Appl. Math 125 (2011) 343-371.
[143]L. Luo and E. G. Fan, Quasi-periodic waves of the N=1 supersymmetric modified Korteweg-de Vries equation, Nonlinear Analysis 74 (2011) 666-6751.
[144]M. Chaichian and P.P. Kulish, On the method of inverse scattering problem and Backlund transformations for supersymmetric equations, Phys. Leet. B 78 (1978) 413-416.
[145]P. H. M. Kersten and P. K. H. Gragert, Symmetries for the super-KdV equation, J. Phys. A:Math. Gen.21 (1988) L579-L584.
[146]Q. P. Liu and M. Manas, Crum transformation and Wronskian type solutions for supersymmetric KdV equation, Phys. Lett. B 396 (1997) 133-140.
[147]Q. P. Liu and M. Manas, Pfaffian solutions for the Manin-Radul-Mathieu SUSY KdV and SUSY sine-Gordon equations, Phys. Lett. B 436 (1998) 306-310.
[148]W. X. Ma, J. S. He and Z. Y. Qin, A supertrace identity and its applications to superintegrable systems, J. Math. Phys.49 (2008) 033511.
[149]S. P. Novikov, A periodic problem for the Korteweg-de Vries equation, Funct. Anal. Appl.8 (1974) 236-246.
[150]B. A. Dubrovin and S. P. Novikov, Periodic and conditionally periodic analogues of the muti-soliton solutions of the Korteweg-de Vries equation, Sov. Phys. JETP 40 (1974) 1058-1063.
[151]B. A. Dubrovin, V. B. Matveev and S. P. Novikov, Non-linear equations of Ko-rteweg-de Vries type, finite-zone linear operator, and Abelian varieties, Russ. Math. Surv.32 (1976) 59-146.
[152]B. A. Dubrovin, Periodic problems for the Korteweg-de Vries equation in the class of finite-zone potentials, Funct. Anal. Appl.9 (1975) 215-223.
[153]I. M. Krichever, An algebraic-geometric construction of the Zaharov-Sabat equations and their periodic solutions, Dokl. Akad. Nauk SSSR 227 (1976) 394-397.
[154]I. M. Krichever, Methods of algebraic geometry in the theory of non-linear equations, Russ. Math. Surv.32 (1977) 185-213.
[155]I. M. Krichever, Integration of nonlinear equations by the methods of algebraic geometry, Funct. Anal. Appl.11 (1977) 12-26.
[156]A. R. Its, and V. B. Matveev, Schrodinger operators with a finite-zone spectrum and the N-soliton solutions of the Korteweg-de Vries equation, Teoret. Mat. Fiz.23 (1975) 51-67.
[157]B. A. Dubrovin, Theta functions and nonlinear equations, Russ. Math. Surv.36 (1981) 11-80.
[158]V. B. Matveev and M. I. Yavor, Solutions presque periodiques et a N-solitons de l'equation hydrodynamique non lineaire de Kaup., Ann. Inst. H Poincare A 31 (1979) 25-41.
[159]E. D. Belokolos, A. I. Bobenko, V. Z. Enol'skii, A. R. Its, and V. B. Matveev, Algebro-Geometric Approach to Nonlinear Integrable Equations, (Springer, Berlin, 1994).
[160]B. A. Dubrovin, Matrix finite-gap operators, Revs. Sci. Tech.23 (1983) 33-78.
[161]E. Date and S. Tanaka, Periodic multi-soliton solutions of Korteweg-de Vries equa-tion and Toda lattice, Progr. Theoret. Phys. Suppl.59 (1976) 107-125.
[162]Y. C. Ma and M. J. Ablowitz, The periodic cubic Schrodinger equation, Stud. Appl. Math.65 (1981) 113-158.
[163]A. O. Smirnov, Real finite-gap regular solutions of the Kaup-Boussinesq equation, Theor. Math. Phys.66 (1986) 19-31.
[164]A. O. Smirnov, Two-gap elliptic solutions to integrable nonlinear equations, Math-ematical Notes 58 (1995) 735-743.
[165]P. D. Lax, Periodic solutions of Korteweg-de Vries equation, Comm. Pure Appl. Math.28 (1975) 141-188.
[166]H. Airault, H. P. McKean and J. Moser, Rational and elliptic solutions of the Korteweg-de Vries equation a related many-body problem, Commun. Pure Appl. Math.30 (1977) 95-148.
[167]H. P. McKean and P. van Moerbeke, The spectrum of Hill's equation, Invent. Math. 30 (1975) 217-274.
[168]H. P. McKean, Integrable Systems and Algebraic Curves, in Global Analysis, eds. M. Grmela and J.E. Marsden, Lecture Notes in Math.755, Springer, Berlin,1979, pp.83-200.
[169]Y. V. Brezhnev, Finite-band potentials with trigonal curves Theor. Math. Phys.133 (2002) 1657-1662
[170]S. Baldwin, J. C. Eilbeck, J. Gibbonsa, and Y. Onishi, Abelian functions for cyclic trigonal curves of genus 4, J. Geo. Phys.58 (2008)450-467
[171]J. C. Eilbeck, V. Z. Enolskii, S. Matsutant, Y. Onishi, and E. Previato, Abelian functions for purely trigonal curves of genus three, Int. Math. Res. Not. IMRN no.1 Art. ID rnm 140 38 pp.2008
[172]Y. Onishi, Abelian functions for trigonal curves of degree four and determinantal formulae in purely trigonal case, Internat. J. Math.20 (2009)427-441.
[173]F. Gesztesy and R. Ratneseelan, An alternative approach to algebro-geometric so-lutions of the AKNS hierarchy, Rev. Math. Phys.10 (1998) 345-391.
[174]F. Gesztesy and H. Holden, Soliton Equations and Their Algebro-Geometric Solu-tions, (Cambridge University Press, Cambridge,2003).
[175]F. Gesztesy and H. Holden, Algebro-geometric solutions of the Camassa-Holm hier-archy, Rev. Mat. Iberoamericana,19 (2003) 73-142.
[176]J. S. Geronimo, F. Gesztesy, and H. Holden, Algebro-geometric solutions of the Baxter-Szego difference equation, Comm. Math. Phys.258 (2005) 149-177.
[177]F. Gesztesy and H. Holden, Real-valued algebro-geometric solutions of the Camassa-Holm hierarchy, Philos. Trans. Roy. Soc. Lond. Ser. A Math. Phys. Eng. Sci.366 (2008) 1025-1054.
[178]F. Gesztesy and R. Weikard, A characterization of all elliptic algebro-geometric solutions of the AKNS hierarchy, Acta Math.181 (1998) 63-108.
[179]F. Gesztesy and R. Weikard, Elliptic algebro-geometric solutions of the KdV and AKNS hierarchies-an analytic approach, Bull. Amer. Math. Soc.35 (1998) 271-317.
[180]R. Dickson, F. Gesztesy, and K. Unterkofler, A new approach to the Boussinesq hierarchy, Math. Nachr.198 (1999) 51-108.
[181]R. Dickson, F. Gesztesy, and K. Unterkofler, Algebro-geometric solutions of the Boussinesq hierarchy, Rev. Math. Phys.11 (1999) 823-879.
[182]F. Gesztesy and H. Holden. A combined sine-Gordon and modified Korteweg-de Vries hierarchy and its algebro-geometric solutions, In:Differential equations and mathematical physics,pp 133-173, Birmingham, AL,1999.
[183]D. J. Kaup and A. C. Newell, An exact solution for a derivative nonlinear Schrodinger equation, J. Math. Phys.19 (1978) 798-801.
[184]V. B. Matveev and A. O. Smirnov, On the Riemann theta function of a trigonal curve and solutions of the Boussinesq and KP equations, Lett. Math. Phys.14 (1987) 25-31.
[185]V. B. Matveev and A. O. Smirnov, Simplest trigonal solutions of the Boussinesq and Kadomtsev-Petviashvili equations, Sov. Phys. Dokl.32 (1987) 202-204.
[186]V. B. Matveev and A. O. Smirnov, Symmetric reductions of the Riemann-function and some of their applications to the Schrodinger and Boussinesq equations, Amer. Math. Soc. Transl.157 (1993) 227-237.
[187]E. Previato, The Calogero-Moser-Krichever system and elliptic Boussinesq solitons, In Hamiltonian Systems, Transformation Groups and Spectral Transform Methods, eds. Harnad J, Marsden J E, CRM, Monreal, pp 57-67,1990.
[188]E. Previato, Monodromy of Boussinesq elliptic operators, Acta Applicandae Math. 36 (1994) 49-55.
[189]E. Previato and J. L. Verdier, Boussinesq elliptic solitons:the cyclic case, In Proc. Indo-French Conf. on Geometry, Dehli, eds. Ramanan S, Beauville A, Hindustan Book Agency, Delhi, pp 173-185,1993.
[190]A. O. Smirnov, A matrix analogue of Appell's theorem and reductions of multidi-mensional Riemann theta-functions, Math. USSR Sbornik 61 (1988) 379-388.
[191]V. E. Zakharov, On stochastization of one-dimensional chains of nonlinear oscillators, Sov. Phys. JETP 38 (1974) 108-110.
[192]H. P. McKean, Boussinesq's equation on the circle, Commun. Pure Appl. Math.34 (1981) 599-691.
[193]H. P. McKean, Boussinesq's equation:How it blows up, in J. C. Maxwell, the Sesquiecentennial Symposium, ed. M.S. Berger, North-Holland, Amsterdam,1984, pp.91-105.
[194]P. Deift, C. Tomei, and E. Trubowitz, Inverse scattering and the Boussinesq equa-tion, Commun. Pure Appl. Math.35 (1982) 567-628.
[195]R. Beals, P. Deift, and C. Tomei, Direct and Inverse Scattering on the Line, (Amer. Math. Soc. Providence, RI,1988).
[196]A. O. Smirnov, On a class of elliptic solutions of the Boussinesq equations, Theoret. Math. Phys.109 (1996) 1515-1522.
[197]A.P. Fordy, J. Gibbons, Factorization of operators Ⅱ, J. Math. Phys.22 (1981) 1170-1175.
[198]P. Griffiths and J. Harris, Principles of Algebraic Geometry, (Wiley, New York, 1994).
[199]D. Mumford, Tata lectures on theta Ⅱ, (Birkhauser, Boston,1984).
[200]H.M. Farkas and I. Kra, Riemann Surfaces,2nd ed., (Springer, New York,1992).
[201]D. J. Kaup, On the inverse scattering problem for cubic eigenvalue problems of the class ψxxx+6qψx+6rψ= λψ, Stud. Appl. Math.62 (1980) 189-216.
[202]W. Hereman and A. Nuseir, Symbolic methods to construct exact solutions of non-linear partial differential equations, Math. Comput. Simul.43 (1997) 13-27.
[203]E. Date, M. Jimbo, M. Kashiwara, and T. Miwa, Transformation groups for soliton equations. Ⅵ. KP hierarchies of orthogonal and symplectic type, J. Phys. Soc. Jpn. 50 (1981) 3813-3818.
[204]S.B. Leble and N. V. Ustinov, Third order spectral problems:reductions and Dar-boux transformations, Inverse Problems 10 (1994) 617-633.
[205]M. Musette and R. Conte, Backlund transformation of partial differential equations from the Painleve-Gambier classification. Ⅰ. Kaup-Kupershmidt equation, J. Math. Phys.39 (1998) 5617-5630.
[206]A. Parker, On soliton solutions of the Kaup-Kupershmidt equation. Ⅰ:Direct bilin-earisation and solitary wave, Physica D 137 (2000) 25-33.
[207]A. Parker, On soliton solutions of the Kaup-Kupershmidt equation. Ⅱ:Anomalous N-soliton solutions, Physica D 137 (2000) 34-48.
[208]M. Musette and C. Verhoeven, Nonlinear superposition formula for the Kaup-Kupershmidt partial differential equation, Physica D 144 (2000) 211-220.
[209]E.G. Reyes, Nonlocal symmetries and the Kaup-Kupershmidt equation, J. Math. Phys.46 (2005) 073507.
[210]E.G. Reyes and G. Sanchez, Explicit solutions to the Kaup-Kupershmidt equation via nonlocal symmetries Internat. J. Bifur. Chaos Appl. Sci. Engrg.17 (2007) 2749-2763.
[2]J. S. Russell, Report on waves, Report of the 14th Meeting of British Association for the Advancement of Science, John Murray, London (1844) 311-390.
[3]J. Boussinesq, Theorie de lintumescence liquid appelee onde solitaire ou de teansla-tion, se propagente dans un canal retangulaire, Compte Rendus Acad. Sci. Paris,72 (1871) 755-759.
[4]J. Boussinesq, Theorie des ondes et de remous qui se propagent le long dun canal rect-angulaire horizontal, en communiquant au liquide contenu dans ce canal des vitesses sensiblemant parielles de la surface au fond, J. Math. Pure Appl. Ser.2,17 (1872) 55-108.
[5]D. J. Korteweg and G. de Vries, On the change of form of long waves advancing in a rectangular canal, an on a new type of long stationary wave, Philos. Mag. Ser.39 (1895) 422-443.
[6]E. Fermi, J. Pasta and S.Ulam, Studies of nonlinear problems, Los Alamos REP. LA (1955) 143-156.
[7]M. Toda, Vibration of chain with nonlinear interaction, J. Phys. Soc. Jpn.22 (1967) 431-436.
[8]N. J. Zabusky and M. D. Kruskal, Interaction of solitons in a collisionless plasma and the recurrence of initial status, Phys. Rev. Lett.15 (1965) 240-243.
[9]C. S. Gardner, J. M. Greene, M. D. Kruskal and R. M. Miura, Method for solving the Korteweg-de Vries equation, Phys. Rev. Lett.19 (1967) 1095-1097.
[10]C. S. Gardner, J. M. Greene, M. D. Kruskal and R. M. Miura, The Korteweg de Vries equation and generalizations. VI. Methods for exact solution, Comm. Pure Appl. Math.21 (1974) 97-133.
[11]M. J. Ablowitz, D.J. Kaup, A.C. Newell and H. Segur, The inverse scattering transform-Fourier analysis foe nonlinear problems, Stud. Appl. Math.53 (1974) 249-315.
[12]M. J. Ablowitz and H. Segur, Solitons and the inverse scattering transform (PA: SIAM, Philadelphia,1981).
[13]M. J. Ablowitz and P. A. Clarkson, Solitons, nonlinear evolution equations and in-verse scattering, London Mathematical Society Lecture Notes Series 149, (Cambridge University Press 2000).
[14]S. Novikov, S. V. Manakov, L. P. Pitaevskii and V. E. Zakharov, Theory ofsolitons, the inverse scattering methods (Consultants Bureau, New York,1984).
[15]R. Beals and R. R. Coifman, Inverse scattering and evolution equations, Commun. Pure Appl. Math.38 (1985) 29-42.
[16]H. Flaschka, On the Toda lattice II. Inverse scattering solutions, Prog. Theor. Phys. 51 (1974) 703-716.
[17]S. V. Manakov, Complete integrability and stochastization of discrete dynamical sys-tems, Sov. Phys. JETP Lett.40 (1975) 269-274.
[18]B. B. Kadomtsev and V. I. Petviashvili, On the stability of solitary waves in weakly dispersive media, Sov. Phys. Dokl.15 (1970) 539-541.
[19]A. Davey and K. Stewartson, On three-dimensional packets of surface waves, Proc. Roy. Soc. London Ser. A 338 (1974) 101-110.
[20]V. E. Zakharov and A. B. Shabat, A scheme for integrating the nonlinear equations of mathematical physics by the method of the inverse scattering problem. I, Funct. Anal. Appl.8 (1974) 226-235.
[21]V. E. Zakharov and A. B. Shabat, Integration of nonlinear equations of mathematical physics by the method of inverse scattering. Ⅱ, Funct. Anal. Appl.13 (1979) 166-174.
[22]P. D. Lax, Integrals of nonlinear equations of evolution and solitary waves, Commun. Pure Appl. Math.21 (1968) 467-490.
[23]R. Hirota, Exact solution of the sine-Gordon equation for mutiple collisions of solitons, J. Phys. Soc. Japan 33 (1972) 1459-1463.
[24]R. Hirota, Exact solution of the Korteweg-de Vries equation for mutiple collisions of solitons, Phys. Rev. Lett.27 (1971) 1192-1194.
[25]R. Hirota, The direct method in soliton theory (Cambridge University Press, Cam-bridge,2004).
[26]X. B. Hu, Nonlinear superposition formula of the Novikov-Veselov equation, J. Phys. A:Math. Gen.27 (1994) 1331-1338.
[27]X. B. Hu, Nonliear superposition formulate for the differential-difference analogue of the KdV equation and two-dimensional Toda equation, J. Phys. A:Math. Gen.27 (1994) 201-214.
[28]X. B. Hu, The higher-order KdV equation with a source and nonlinear superposition formula, Chaos Solitons and Fractals 7 (1996) 211-215.
[29]D. J. Zhang, The N-soliton solutions for the modified KdV equation with self-consistent sources, J. Phys. Soc. Jpn.71 (2002) 2649-2656.
[30]D. J. Zhang and D. Y. Chen, The N-soliton solutions of the sine-Gordon equation with self-consistent sources, Physica A 321 (2003) 467-481.
[31]J. Hietarinta and D. J. Zhang, Multisoliton solutions to the lattice Boussinesq equa-tion, J. Math. Phys.51 (2010) 033505.
[32]V. B. Matveev and M. A. Salle, Darboux transformations and solitons (Springer,
[33]谷超豪,胡和生,周子翔,孤立子理论中的达布变换及其几何应用(上海科学技术出版社,上海,1999).
[34]X. G. Geng and H. W. Tam, Darboux transformation and soliton solutions for gen-eralized nonlinear Schrodinger equations, J. Phys. Soc. Jpn.68 (1999) 1508-1512.
[35]E. G. Fan, Darboux transformation for soliton-like solutions for the Gerdjikov-Ivanov equatiion, J. Phys. A:Math. Gen.33 (2000) 6925-6933.
[36]Y. S. Li and J. E. Zhang, Darboux transformations of classical Boussonesq syetem and its new solutions, Phys. Lett. A.284 (2001) 253-258.
[37]C. Rogers and W. K. Schief, Backlund and Darboux transformations geometry and modern applcations in soliton theory, (Cambridge University Press, Cambridge, 2002).
[38]M. Bordemann, M. forger, J. Laartz and U. Schaper, The Lie-Poisson structure of integrable classical nonlinear sigma models, Comm. Math. Phys.152 (1993) 167-190.
[39]B. Jovanovic, Non-holonomic geodesic flows on Lie groups and the integrable Suslov problem on SO(4), J. Phys. A:Math. Gen.31 (1998) 1415-1422.
[40]R. Z. Zhdanov and V. I. Lahno, Group classification of heat conductivity equations with a nonlinear source, J. Phys. A:Math. Gen.32 (1999) 7405-7418.
[41]S. Spichak and V. Stognii, Symmetry classification and exact solutions of the one-dimensional Fokker-Planck equation with arbitrary coefficients of drift and diffusion, J. Phys. A:Math. Gen.32 (1999) 8341-8353.
[42]D. L. Du, Complex form, reduction and Lie-Poisson structure for the nonlinearized spectral problem of the Heisenberg hierarchy, Physica A 303 (2001) 439-456.
[43]C. W. Cao, Nonlinearization of the Lax system for AKNS hierarchy, Science China A 33 (1990) 528-536.
[44]C. W. Cao and X. G. Geng, C Neumann and Bargmann systems associated with the coupled KdV soliton hierarchy, J. Phys. A:Math. Gen.23 (1990) 4117-4125.
[45]C. W. Cao, A classical integrable system and the involutive representation of solutions of the KdV equation, Acta. Math. Sinica 7 (1991) 216-223.
[46]C. W. Cao and X. G. Geng, A nonconfocal generator of involutive systems and three associated soliton hierarchies, J. Math. Phys.32 (1991) 2323-2328.
[47]X. G. Geng, Discrete Bargmann and Neumann systems and finite-dimensional inte-grable systems, Physica A 212 (1994) 132-142.
[48]C. W. Cao, X. G. Geng and Y. T. Wu, From the special (2+1)-Toda lattice to the Kadomtsev-Petviashvili equation, J. Phys. A:Math. Gen.32 (1999) 8059-8078.
[49]X. G. Geng and H. H. Dai, Nonlinearization of the eigenvalue problem related to coupled nonlinear Schrodinger equations, J. Math. Anal. Appl.233 (1999) 26-55.
[50]X. G. Geng and C. W. Cao, Decomposition of the (2+1)-dimensional Gardner equa-tion and its quasi-periodic solutions, Nonlinearity 14 (2001) 1433-1452.
[51]R. G. Zhou, Nonlinearizations of spectral problems of the nonlinear Schrodinger equa-tion and the real-valued modified Korteweg-de Vries equation, J. Math. Phys.48 (2007) 013510.
[52]R. G. Zhou, Relationships among three kinds of finite dimensional integrable Hamil-tonian systems, Physica A 320 (2003) 128-140.
[53]C. W. Cao, Parametric representation of the finite-band solution of the Heisenberg equation, Phys. Lett. A.184 (1994) 333-338.
[54]X. G. Geng, C. W. Cao and H. H. Dai, Quasi-periodic solutions for some (2+1)-dimensional integrable models generated by the Jaulent-Miodek hierarchy, J. Phys. A:Math. Gen.34 (2001) 989-1004.
[55]C. W. Cao, X. G. Geng and H. Y. Wang, Algebraic-geometric solution of the (2+1)-dimensional Burgers equatiion with a discrete variable, J. Math. Phys.43 (2002) 621-623.
[56]C. W. Cao, Y. T. Wu, and X. G. Geng, Relation between the Kadomtsev-Petviashvili equation and the Confocal involutive system, J. Math. Phys.40 (1999) 3948-3970.
[57]X. G. Geng, Y. T. Wu and C. W. Cao, Quasi-periodic solutions of the modified Kadomtsev-Petviashvili equation, J. Phys. A:Math. Gen.32 (1999) 3733-3742.
[58]X. G. Geng and Y. T. Wu, Finite-band solutions of the classical Boussinesq-Burgers equations, J. Math. Phys.40 (1999) 2971-2982.
[59]X. G. Geng, H. H. Dai and C. W. Cao, Algebro-geometric constructions of the discrete Ablowitz-Ladik flows and applications, J. Math. Phys.44 (2003) 4573-4588.
[60]X. G. Geng, Algebraic-geometrical solutions of some multidimensional nonlinear evo-lution equations, J. Phys. A:Math. Gen.36 (2003) 2289-2303.
[61]H. H. Dai and X. G. Geng, Explicit solutions of the (2+1)-dimensional modified Toda Lattice through straightening out of the relativistic Toda flows, J. Phys. Soc. Jpn.72 (2003) 3063-3069.
[62]X. G. Geng, H. H. Dai, and J. Y. Zhu, Decomposition of the discrete Ablowitz-Ladik hierarchy, Stud. Appl. Math.118 (2007) 281-312.
[63]Z. J. Qiao, Generalized r-matrix structure and algebro-geometric solution for inte-grable systems, Rev. Math. Phys.21 (2001) 545-586.
[64]Z. J. Qiao, The Camassa-Holm hierarchy, N-dimensional integrable systems, and algebro-geometric solution on a symplectic submanifold, Comm. Math. Phys.239 (2003) 309-341.
[65]V. E. Zakharov and S. V. Manakov, Generalization of the method of inverse scattering problem, Teoret. Mat. Fiz.27 (1976) 283-287.
[66]E. A. Kuznetsov, M. D. Spector and G. E. Falkovich, On the stability of nonlinear waves in integrable models, Physica D 10 (1984) 379-386.
[67]A. I. Zenchuk, A unified dressing method for C-integrable hierarchies; the particular example of a (3+1)-dimensional n-wave equation, J. Phys. A:Math. Gen.37 (2004) 6557-6571.
[68]H. H. Dai and A. Jeffery, The inverse scattering transforms for certain types of variable coefficient KdV equations, Phys. Lett. A 139 (1989) 369-372.
[69]A. Jeffery and H. H. Dai, On the application of a generalized version of the dressing method to the integration of variable coefficient KdV equation, Rendiconti di Math., Serie VII 10(1990) 439-455.
[70]J. Y. Zhu and X. G. Geng, The generalized version of dressing method with applica-tions to AKNS equations, J. Nonlinear Math. Phys.13 (2006) 81-89.
[71]M. J. Ablowitz, A. Ramani and H. Segur, A connection between nonlinear evolution equations and ordinary diefrential equations of p-type I, J. Math. Phys.21 (1980) 715-721.
[72]M. J. Ablowitz, A. Ramani and H. Segur, A connection between nonlinear evolution equations and ordinary diefrential equations of p-type II, J. Math. Phys.21 (1980) 1006-1015.
[73]J. B. McLeod and P. J. Olver, The connection between patial differential equations soluble by inverse scattering and odinary dieferntial equations of painleve type, J. Math. Anal.14 (1983) 488-506.
[74]R. Conte, Invariant painleve analysis of Partial difereniial equations, Phys. Lett. A 140 (1989) 383-390.
[75]S. Y. Lou and Q. X. Wu, Painleve integrability of two sets of nonlinear evolution equations with nonlinear dispersions, Phys. Lett. A 262 (1999) 344-349.
[76]S. L. Zhang, B. Wu and S. Y. Lou, Painleve analysis and special solutions of gener-alized Broer-Kaup equations, Phys. Lett. A.300 (2002) 40-48.
[77]Y. Cheng and Y. S. Li, The constraint of the Kadometsev-Petviashvili equation and its special solutions, Phys. Lett. A 157 (1991) 22-26.
[78]Y. Cheng and Y. S. Li, Constraints of the (2+l)-dimensional integrable soilton sys-tems, J. Phys. A:Math. Gen.25 (1992) 419-431.
[79]S. Y. Lou and J. Lin, The generalized symmetry algebraic of the bilinear Kadomtsev-Petviashvili equation, Phys. Lett. A 185 (1994) 29-34.
[80]S. Y. Lou and X. B. Hu, Infinitely many Lax pairs and symmetry constraints of the KP equation, J. Math. Phys.38 (1997) 6401-6427.
[81]C. Z. Qu, Symmetries and solutions to the thin film equations, J. Math. Anal. Appl. 317 (2006) 381-397.
[82]C. Z. Qu, S. L. Zhang and R. C. Liu, Separation of variables and exact solutions to quasilinear diffusion equations with nonlinear source, Physica D 144 (2000) 97-123.
[83]P. G. Estevez, C. Z. Qu and S. L. Zhang, Separation of variables of a generalized porous medium equation with nonlinear source, J. Math. Anal. Appl.275 (2002) 44-59.
[84]J. Y. Hu and C. Z. Qu, Functionally separable solutions to nonlinear wave equations by group foliation method, J. Math. Anal. Appl.330 (2007) 298-311.
[85]M. L. Wang, Exact solutions for a compound KdV-Burgers equation, Phys. Lett. A 213 (1996) 279-287.
[86]M. L. Wang and Y. M. Wang, A new Backlund transformation and multi-solutions to the KdV equation with general variable coefficients, Phys. Lett. A 287 (2001) 211-216.
[87]R. G. Zhou, The finite-band solutions of the Jaulent-Miodek equation, J. Math. Phys. 38 (1997) 2535-2546.
[88]R. G. Zhou, Integrable Rosochatius deformations of the restricted soliton flows, J. Math. Phys.48 (2007) 2535-2551.
[89]Z. J. Qiao, G. P. Zhang, on peaked and smooth solitons for the Camassa-Holm equa-tion, Europhys. Lett.73 (2006) 657-663.
[90]Z. J. Qiao, The matrix and an algebraic-geometric solution of the AKNS system, Theore. Math. Phys.127 (2001) 827-834.
[91]J. Wess and B. Zumino, Supergauge transformations in four dimensions, Nucl. Phys, B 70 (1974) 39-50.
[92]V. G. Kac, A Sketch of Lie Superalgebra Theory, Commun. Math. Phys.53 (1977) 31-64.
[93]V. G. Kac, Lie super algebras, Adv. Math.26 (1977) 8-96.
[94]F. A. Berezin and D. A. Leites, Supermanifolds, Dokl. Akad. Nauk SSSR 224 (1975) 505-508.
[95]V. Rittenberg, A guide to Lie superalgebras, Group Theoretical Methods in Physics 79 (1978) 3-21.
[96]D. A. Leites, Lie superalgebras, Funct. Anal.9 (1975) 75-76.
[97]D. A. Leites, Introduction to the theory of supermanifolds, Russ. Math. Surv.35 (1980) 1-64.
[98]F. Berezin, Introduction to super analysis, (D. Reidel Publishing Company, Dor-drecht,1987).
[99]B. S. Dewitt, Supermanifolds (Cambrige University Press,1992).
[100]F. A. Rogers, Supermanifolds:theorey and application, (World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ,2007).
[101]B. A. Kupershmidt, A super Korteweg-de Vries equation:an integrable system, Phys. Lett. A 102 (1984) 213-215.
[102]B. A. Kupershmidt, Bosons and fermions interacting integrably with the Korteweg-de Vries field, J. Phys. A:Math. Gen.17 (1984) L869-L872.
[103]B. A. Kupershmidt, Super Korteweg-de Vries equation associated to super extension of virasoro algebra, Phys. Lett. A 109 (1985) 417-423.
[104]M. Gurses and O. Oguz, A super AKNS scheme, Phys. Lett. A 108 (1985) 437-440.
[105]M. Gurses and O. Oguz, A super soliton connection, Lett. Math. Phys.11 (1986) 235-246.
[106]Y. S. Li and L. N. Zhang, Super AKNS scheme and its infinite conserved currents, Nuovo Cimento A 93 (1986) 175-183.
[107]Y. S. Li and L. N. Zhang, A note on the super AKNS equations, J. Phys. A:Math. Gen.21 (1988) 1549-1552.
[108]Y. S. Li and L. N. Zhang, Hamiltonian structure of the super evolution equation, J. Math. Phys.31 (1990) 470-475.
[109]X. B. Hu, An approach to generate superextentions of integrable systems, J. Phys. A:Math. Gen.30 (1997) 619-632.
[110]M. Antonowicz and A. P. Fordy, Super-Extensions of Energy Dependent Sehrodinger Operators, Commun. Math. Phys.124 (1989) 487-500.
[Ill]Yu. I. Manin and A.0. Radul, A supersymmetric extension of the Kadomtsev-Petviashvili hierarchy, Commun. Math. Phys.98 (1985) 65-77.
[112]P. Mathieu, Supersymmetric extension of the Korteweg-de Vries Equation, J. Math. Phys.29 (1988) 2499-2506.
[113]Z. Popowicz, The fully supersymmetric AKNS equations, J. Phys. A:Math. Gen. 23 (1990) 1127-1136.
[114]Z. Popowicz, A 2-component or N= 2 supersymmetric Camassa-Holm equation, Phys. Lett. A 354 (2006) 110-114.
[115]Z. Popowicz, Odd Hamiltonian structure for supersymmetric Sawada-Kotera equa-tion, Phys. Lett. A 373 (2009) 3315-3323.
[116]Q. P. Liu, Supersymmetric Harry-Dym type equations, J. Phys. A:Math. Gen.28 (1995) L245-L248.
[117]Q. P. Liu, Ziemowit Popowicz and K. Tian, The even and odd supersymmetric Hunter-Saxton and Liouville equations, Phys. Lett. A 375 (2010) 29-35.
[118]K. Tian and Q. P. Liu, A supersymmetric Sawada-Kotera equation, Phys. Lett. A 373 (2009) 1807-1810
[119]Q. P. Liu, Fully supersymmetric hierarchies from an energy-dependent super Hill operator, J. Phys. A:Math. Gen.30 (1997) 8661-8670.
[120]G. H. M. Roelofs and P. H. M. Kersten, Supersymmetric extensions of the nonlinear Sehrodinger equation:Symmetries and coverings, J. Math. Phys.33 (1992) 2185-2206.
[121]S. Q. LU, X. B. Hu and Q. P. Liu, A supersymmetric Ito's equation and its soliton solutions, J. Phys. Soc. Jpn.75 (2006) 064004.
[122]J. C. Beuneli, A. Das snd Z. Popowicz, Supersymmetric extensions of the Harry Dym hierarchy, J. Math. Phys.44 (2003) 4756-4767.
[123]C. M. Yung, The N=2 supersymmetric Boussinesq hierarchies, Phys. Lett. B 309 (1993) 75-84.
[124]J. Lenells and O. Lechtenfeld, On the N=2 supersymmetric Camassa-Holm and Hunter-Saxton equations, J. Math. Phys.50 (2009) 012704.
[125]M. A. Olshanetsky, Supersymmetric Two-Dimensional Toda Lattice, Commun. Math. Phys.88 (1983) 63-76.
[126]K. Ikeda, A Supersymmetric Extension of the Toda Lattice Hierarchy, Lett. Math. Phys.14 (1987) 321-328.
[127]P. Mathieu, Hamiltonian structure of graded and super evolution equations, Lett. Math. Phys.16 (1988) 199-206.
[128]Q. P. Liu, Bi-Hamiltonian structure of a genralized super Korteweg-de Vrie equation, J. Phys. A:Math. Gen.26 (1993) L1239-L1242.
[129]Y. Watanabe, A Hamilton Formalism for Evolution Equations with Odd Variables, Annali di Matematica pura ed applicata (IV), Vol. CLIV (1989) 127-146.
[130]W. Oevel and Z. Popowicz, The Bi-Hamiltonian Structure of Fully Supersymmetric Korteweg-de Vries Systems, Commun. Math. Phys.139 (1991) 441-460.
[131]J. F. Gomes, L. H. Ymai, A. H. Zimerman, Permutability of Backlund transforma-tion for N=1 supersymmetric sinh-Gordon, Phys. Lett. A 373 (2009) 1401-1404.
[132]A. R. Chowdhury and S. Roy, On the Backlund transformation and Hamiltonian properties of superevaluation equations, J. Math. Phys.27 (1986) 2464-2468.
[133]Q. P. Liu, Darboux transformations for supersymmetric Korteweg-de Vries equa-tions, Lett. Math. Phys.35 (1995) 115-122.
[134]Q. P. Liu and M. Manas, Darboux transformation for the Manin-Radul supersym-metric KdV equation, Phys. Lett. B 394 (1997) 337-342.
[135]Q. P. Liu and M. Manas, Darboux transformation for supersymmetric Kadomtsev-Petviashvili hierarchy, Phys. Lett. B 485 (2000) 293-300.
[136]M. Siddiq, M. Hassan and U. Saleem, On Darboux transformation of the supersym-metric sine-Gordon equation, J. Phys. A:Math.Gen.39 (2006) 7313-7318.
[137]A. S. Carstea, Extension of the bilinear formalism to supersymmetric KdV-type equations, Nonlinearity 13 (2000) 1645-1656.
[138]A. S. Carstea, Bilinear approach to supersymmetric KdV equation, J. Nonlinear Math. Phys.8 (Suppl.) (2001) 48-52.
[139]Q. P. Liu and X. B. Hu, Biliearization of N= 1 supersymmetric Korteweg-de Vries equation revisited, J. Phys. A:Math. Gen.38 (2005) 6371-6378.
[140]Q. P. Liu, X. B. Hu and M. X. Zhang, Supersymmetric modified Korteweg-de Vries equation:bilinear approach, Nonlinearity 18 (2005) 1597-1603.
[141]E. G. Fan, Supersymmetric KdV-Sawada-Kotera-Ramani equation and its quasi-periodic wave solutions, Phys. Lett. A 374 (2010) 744-749.
[142]E. G. Fan and Y. C. Hon, Quasiperiodic Wave Solutions of N=2 Supersymmetric KdV Equation in Superspace, Stud. Appl. Math 125 (2011) 343-371.
[143]L. Luo and E. G. Fan, Quasi-periodic waves of the N=1 supersymmetric modified Korteweg-de Vries equation, Nonlinear Analysis 74 (2011) 666-6751.
[144]M. Chaichian and P.P. Kulish, On the method of inverse scattering problem and Backlund transformations for supersymmetric equations, Phys. Leet. B 78 (1978) 413-416.
[145]P. H. M. Kersten and P. K. H. Gragert, Symmetries for the super-KdV equation, J. Phys. A:Math. Gen.21 (1988) L579-L584.
[146]Q. P. Liu and M. Manas, Crum transformation and Wronskian type solutions for supersymmetric KdV equation, Phys. Lett. B 396 (1997) 133-140.
[147]Q. P. Liu and M. Manas, Pfaffian solutions for the Manin-Radul-Mathieu SUSY KdV and SUSY sine-Gordon equations, Phys. Lett. B 436 (1998) 306-310.
[148]W. X. Ma, J. S. He and Z. Y. Qin, A supertrace identity and its applications to superintegrable systems, J. Math. Phys.49 (2008) 033511.
[149]S. P. Novikov, A periodic problem for the Korteweg-de Vries equation, Funct. Anal. Appl.8 (1974) 236-246.
[150]B. A. Dubrovin and S. P. Novikov, Periodic and conditionally periodic analogues of the muti-soliton solutions of the Korteweg-de Vries equation, Sov. Phys. JETP 40 (1974) 1058-1063.
[151]B. A. Dubrovin, V. B. Matveev and S. P. Novikov, Non-linear equations of Ko-rteweg-de Vries type, finite-zone linear operator, and Abelian varieties, Russ. Math. Surv.32 (1976) 59-146.
[152]B. A. Dubrovin, Periodic problems for the Korteweg-de Vries equation in the class of finite-zone potentials, Funct. Anal. Appl.9 (1975) 215-223.
[153]I. M. Krichever, An algebraic-geometric construction of the Zaharov-Sabat equations and their periodic solutions, Dokl. Akad. Nauk SSSR 227 (1976) 394-397.
[154]I. M. Krichever, Methods of algebraic geometry in the theory of non-linear equations, Russ. Math. Surv.32 (1977) 185-213.
[155]I. M. Krichever, Integration of nonlinear equations by the methods of algebraic geometry, Funct. Anal. Appl.11 (1977) 12-26.
[156]A. R. Its, and V. B. Matveev, Schrodinger operators with a finite-zone spectrum and the N-soliton solutions of the Korteweg-de Vries equation, Teoret. Mat. Fiz.23 (1975) 51-67.
[157]B. A. Dubrovin, Theta functions and nonlinear equations, Russ. Math. Surv.36 (1981) 11-80.
[158]V. B. Matveev and M. I. Yavor, Solutions presque periodiques et a N-solitons de l'equation hydrodynamique non lineaire de Kaup., Ann. Inst. H Poincare A 31 (1979) 25-41.
[159]E. D. Belokolos, A. I. Bobenko, V. Z. Enol'skii, A. R. Its, and V. B. Matveev, Algebro-Geometric Approach to Nonlinear Integrable Equations, (Springer, Berlin, 1994).
[160]B. A. Dubrovin, Matrix finite-gap operators, Revs. Sci. Tech.23 (1983) 33-78.
[161]E. Date and S. Tanaka, Periodic multi-soliton solutions of Korteweg-de Vries equa-tion and Toda lattice, Progr. Theoret. Phys. Suppl.59 (1976) 107-125.
[162]Y. C. Ma and M. J. Ablowitz, The periodic cubic Schrodinger equation, Stud. Appl. Math.65 (1981) 113-158.
[163]A. O. Smirnov, Real finite-gap regular solutions of the Kaup-Boussinesq equation, Theor. Math. Phys.66 (1986) 19-31.
[164]A. O. Smirnov, Two-gap elliptic solutions to integrable nonlinear equations, Math-ematical Notes 58 (1995) 735-743.
[165]P. D. Lax, Periodic solutions of Korteweg-de Vries equation, Comm. Pure Appl. Math.28 (1975) 141-188.
[166]H. Airault, H. P. McKean and J. Moser, Rational and elliptic solutions of the Korteweg-de Vries equation a related many-body problem, Commun. Pure Appl. Math.30 (1977) 95-148.
[167]H. P. McKean and P. van Moerbeke, The spectrum of Hill's equation, Invent. Math. 30 (1975) 217-274.
[168]H. P. McKean, Integrable Systems and Algebraic Curves, in Global Analysis, eds. M. Grmela and J.E. Marsden, Lecture Notes in Math.755, Springer, Berlin,1979, pp.83-200.
[169]Y. V. Brezhnev, Finite-band potentials with trigonal curves Theor. Math. Phys.133 (2002) 1657-1662
[170]S. Baldwin, J. C. Eilbeck, J. Gibbonsa, and Y. Onishi, Abelian functions for cyclic trigonal curves of genus 4, J. Geo. Phys.58 (2008)450-467
[171]J. C. Eilbeck, V. Z. Enolskii, S. Matsutant, Y. Onishi, and E. Previato, Abelian functions for purely trigonal curves of genus three, Int. Math. Res. Not. IMRN no.1 Art. ID rnm 140 38 pp.2008
[172]Y. Onishi, Abelian functions for trigonal curves of degree four and determinantal formulae in purely trigonal case, Internat. J. Math.20 (2009)427-441.
[173]F. Gesztesy and R. Ratneseelan, An alternative approach to algebro-geometric so-lutions of the AKNS hierarchy, Rev. Math. Phys.10 (1998) 345-391.
[174]F. Gesztesy and H. Holden, Soliton Equations and Their Algebro-Geometric Solu-tions, (Cambridge University Press, Cambridge,2003).
[175]F. Gesztesy and H. Holden, Algebro-geometric solutions of the Camassa-Holm hier-archy, Rev. Mat. Iberoamericana,19 (2003) 73-142.
[176]J. S. Geronimo, F. Gesztesy, and H. Holden, Algebro-geometric solutions of the Baxter-Szego difference equation, Comm. Math. Phys.258 (2005) 149-177.
[177]F. Gesztesy and H. Holden, Real-valued algebro-geometric solutions of the Camassa-Holm hierarchy, Philos. Trans. Roy. Soc. Lond. Ser. A Math. Phys. Eng. Sci.366 (2008) 1025-1054.
[178]F. Gesztesy and R. Weikard, A characterization of all elliptic algebro-geometric solutions of the AKNS hierarchy, Acta Math.181 (1998) 63-108.
[179]F. Gesztesy and R. Weikard, Elliptic algebro-geometric solutions of the KdV and AKNS hierarchies-an analytic approach, Bull. Amer. Math. Soc.35 (1998) 271-317.
[180]R. Dickson, F. Gesztesy, and K. Unterkofler, A new approach to the Boussinesq hierarchy, Math. Nachr.198 (1999) 51-108.
[181]R. Dickson, F. Gesztesy, and K. Unterkofler, Algebro-geometric solutions of the Boussinesq hierarchy, Rev. Math. Phys.11 (1999) 823-879.
[182]F. Gesztesy and H. Holden. A combined sine-Gordon and modified Korteweg-de Vries hierarchy and its algebro-geometric solutions, In:Differential equations and mathematical physics,pp 133-173, Birmingham, AL,1999.
[183]D. J. Kaup and A. C. Newell, An exact solution for a derivative nonlinear Schrodinger equation, J. Math. Phys.19 (1978) 798-801.
[184]V. B. Matveev and A. O. Smirnov, On the Riemann theta function of a trigonal curve and solutions of the Boussinesq and KP equations, Lett. Math. Phys.14 (1987) 25-31.
[185]V. B. Matveev and A. O. Smirnov, Simplest trigonal solutions of the Boussinesq and Kadomtsev-Petviashvili equations, Sov. Phys. Dokl.32 (1987) 202-204.
[186]V. B. Matveev and A. O. Smirnov, Symmetric reductions of the Riemann-function and some of their applications to the Schrodinger and Boussinesq equations, Amer. Math. Soc. Transl.157 (1993) 227-237.
[187]E. Previato, The Calogero-Moser-Krichever system and elliptic Boussinesq solitons, In Hamiltonian Systems, Transformation Groups and Spectral Transform Methods, eds. Harnad J, Marsden J E, CRM, Monreal, pp 57-67,1990.
[188]E. Previato, Monodromy of Boussinesq elliptic operators, Acta Applicandae Math. 36 (1994) 49-55.
[189]E. Previato and J. L. Verdier, Boussinesq elliptic solitons:the cyclic case, In Proc. Indo-French Conf. on Geometry, Dehli, eds. Ramanan S, Beauville A, Hindustan Book Agency, Delhi, pp 173-185,1993.
[190]A. O. Smirnov, A matrix analogue of Appell's theorem and reductions of multidi-mensional Riemann theta-functions, Math. USSR Sbornik 61 (1988) 379-388.
[191]V. E. Zakharov, On stochastization of one-dimensional chains of nonlinear oscillators, Sov. Phys. JETP 38 (1974) 108-110.
[192]H. P. McKean, Boussinesq's equation on the circle, Commun. Pure Appl. Math.34 (1981) 599-691.
[193]H. P. McKean, Boussinesq's equation:How it blows up, in J. C. Maxwell, the Sesquiecentennial Symposium, ed. M.S. Berger, North-Holland, Amsterdam,1984, pp.91-105.
[194]P. Deift, C. Tomei, and E. Trubowitz, Inverse scattering and the Boussinesq equa-tion, Commun. Pure Appl. Math.35 (1982) 567-628.
[195]R. Beals, P. Deift, and C. Tomei, Direct and Inverse Scattering on the Line, (Amer. Math. Soc. Providence, RI,1988).
[196]A. O. Smirnov, On a class of elliptic solutions of the Boussinesq equations, Theoret. Math. Phys.109 (1996) 1515-1522.
[197]A.P. Fordy, J. Gibbons, Factorization of operators Ⅱ, J. Math. Phys.22 (1981) 1170-1175.
[198]P. Griffiths and J. Harris, Principles of Algebraic Geometry, (Wiley, New York, 1994).
[199]D. Mumford, Tata lectures on theta Ⅱ, (Birkhauser, Boston,1984).
[200]H.M. Farkas and I. Kra, Riemann Surfaces,2nd ed., (Springer, New York,1992).
[201]D. J. Kaup, On the inverse scattering problem for cubic eigenvalue problems of the class ψxxx+6qψx+6rψ= λψ, Stud. Appl. Math.62 (1980) 189-216.
[202]W. Hereman and A. Nuseir, Symbolic methods to construct exact solutions of non-linear partial differential equations, Math. Comput. Simul.43 (1997) 13-27.
[203]E. Date, M. Jimbo, M. Kashiwara, and T. Miwa, Transformation groups for soliton equations. Ⅵ. KP hierarchies of orthogonal and symplectic type, J. Phys. Soc. Jpn. 50 (1981) 3813-3818.
[204]S.B. Leble and N. V. Ustinov, Third order spectral problems:reductions and Dar-boux transformations, Inverse Problems 10 (1994) 617-633.
[205]M. Musette and R. Conte, Backlund transformation of partial differential equations from the Painleve-Gambier classification. Ⅰ. Kaup-Kupershmidt equation, J. Math. Phys.39 (1998) 5617-5630.
[206]A. Parker, On soliton solutions of the Kaup-Kupershmidt equation. Ⅰ:Direct bilin-earisation and solitary wave, Physica D 137 (2000) 25-33.
[207]A. Parker, On soliton solutions of the Kaup-Kupershmidt equation. Ⅱ:Anomalous N-soliton solutions, Physica D 137 (2000) 34-48.
[208]M. Musette and C. Verhoeven, Nonlinear superposition formula for the Kaup-Kupershmidt partial differential equation, Physica D 144 (2000) 211-220.
[209]E.G. Reyes, Nonlocal symmetries and the Kaup-Kupershmidt equation, J. Math. Phys.46 (2005) 073507.
[210]E.G. Reyes and G. Sanchez, Explicit solutions to the Kaup-Kupershmidt equation via nonlocal symmetries Internat. J. Bifur. Chaos Appl. Sci. Engrg.17 (2007) 2749-2763.