Landau-Lifshitz方程的反散射方法
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摘要
本论文研究可积Landau-Lifshitz方程的反散射方法.经典可积情形下的Landau-Lifshitz方程早在上个世纪70年代末就已有大量的研究工作.国内的很多学者,在80年代,90年代初期亦研究过此经典模型的相关问题.经典可积的模型,虽然有其理想的地方,但是它为与之相关非可积模型研究提供一些思路.从中发现一些新现象和新问题亦为非可积的方程的研究指明方向.本文的主要工作是运用发展的反散射方法—Riemann-Hilbert方法研究此经典模型.由于近年来反散射方法的发展,利用此方法研究此经典模型,仍然可以得到一些新的结果.首先,我们可以得到在反散射框架下的解的存在唯一性.其次对于离散散射数据和精确解的求解,我们利用的主要工具是推广的Darboux变换.首次将推广的Darboux变换结合反散射方法巧妙地处理反散射方法中多重极点的散射数据的求解问题.据此,我们可以给出一般的孤立子求解公式,进而给孤立子解以完全的分类.除此之外,我们可以运用Deift-Zhou方法分析解的长时间渐近行为.反散射方法作为求解可积偏微分方程柯西问题的重要方法,亦提供了研究微分方程的重要手段.近年来,非线性科学领域内出现了新的研究对象—怪波.它的产生机制为调制不稳定性.此时,由于涉及到微分方程的不适定性,微分方程的一些理论已经无法分析.反散射方法为这些新现象的研究亦提供了重要手段.
第一章为背景介绍.首先介绍方程的背景知识,研究进展.其次介绍反散射方法的历史背景,重要进展和技巧,以及一些最新结果.最后给出了本文的创新点和主要结果.
第二章研究经典Landau-Lifshitz方程的反散射方法.首先运用规范变换和反散射方法适定性理论的新结果,得到Landau-Lifshitz方程在一个带权的Soblev空间的适定性.其次,运用推广的Darboux变换方法将孤立了解给予完全的分类.最后利用Deift-Zhou方法分析一般孤立子解的长时问渐近行为.
第三章研究球对称情形可积的Landau-Lifshitz方程,此模型的可积性研究早在1994年由Lakshamnan等人提出.关于此模型精确解的研究已有些结果,但是利用反散射方法研究该问题尚属首次.主要因为其相应的谱问题为非等谱的半线问题.我们需要对谱问题进行适当的延拓.这里对于奇数维和偶数维系统运用不同的延拓方式.除此之外,我们还对方程解的动力学行为进行分析.同时我们顺便给出了推广的NLS方程的孤子解的动力学行为研究.
第四章研究经典可积Landau-Lifshitz方程在自旋波背景下的反散射问题.运用的主要工具仍然是规范变换和反散射.首先,我们利用规范变换将其变为聚焦的NLS方程在平面波背景下的反散射问题.利用规范变换理论,我们研究其守恒律以及Galilean变换.最后我们利用推广的Darboux变换得到其一般的孤立子解公式.并且具体给出了呼吸子解和怪波解的表达式,同时通过画图分析这些具体解的动力学行为.
In this dissertation, we consider the inverse scattering method of Landau-Lifshitz equation. There are lots of works concerned the integrable Landau-Lifshitz equation in the70s of the last century. Domestic scholars research the relevant issues of this model also in the80s and90s of the last century. Although the classical integrable model is idealization, it provides research idea of the other non-integrable models. Therefore, finding some new phenomenon and new problems in the integrable models can be used to define the research orientation for the non-integrable ones. The main tasks in this dissertation are researching the classical model by inverse scattering method or Riemann-Hilbert approach. On account of the recent developments of inverse scattering method, we can obtain some new results by this approach. Firstly, we can obtain the well-posedness in the frame of inverse scattering. Secondly, we take advantage of the generalized Darboux transformation to tackle with the discrete scattering data and exact solution. We combine generalized Darboux transformation with inverse scattering method for the first time. By this cohesion, we ingeniously deal with the high order pole scattering coefficient. Moreover, the general soliton formula and the complete classification of soliton solution can be obtained. Finally, we can analysis the long time asymptotics of this model by Deift-Zhou method. These results can not obtained by classical PDE analysis. Therefore, the inverse scattering method, as a method to solve the Cauchy problems of integrable partial differential equation, provides the important ways to research it. Recently, there is a hot topic-rogue wave in the nonlinear science. An important physical mechanism to explain the rogue wave is the modulational instability. Owning to the ill-posednees of the model, we can not use the theory of analysis of PDEs in this instance. The inverse scattering method provides new ways to tackle with it.
In chapter1, we briefly introduce the physical background and historical results of the model. Besides, we present the historical background, some new results and important progress with respect to the inverse scattering method. The main innovative points are also presented in this part.
In chapter2, we concern on the inverse scattering method of the classical integrable Landau-Lifshitz equation. Firstly, we obtain the well-posedness in a weighted Soblev space by combining the gauge transformation and some known results. Secondly, we present the completely classification of soliton solution by combining the generalized Darboux transformation and inverse scattering method. Finally, we analyse the long time asymptotics of Landau-Lifshitz equation by Deift-Zhou nonlinear steepest descent method.
In chapter3, we concern on the spherical symmetry Landau-Lifshitz (ssL-L) equa-tion. The integrability of this model is given in by Lakshamnan group. There are some results about exact solution. But there is no result about the Cauchy problem about this model, because it is the non-isospectral half line problem. We need extend the spectral problem on the line properly. Besides, we analyse the dynamics of solution of ssL-L equation. As a by-product, we show the dynamics of the generalized NLS equation.
In chapter4, we consider the inverse scattering problem of Landau-Lifshitz equa-tion on the spin wave background. We give the gauge transformation and inverse scattering analysis. Firstly, we take advantage of gauge transformation to convert this problem into focusing NLS equation on the plane wave background. Besides, we give the Galilean transformation to Landau-Lifshitz equation and conversation laws. Finally, we utilize the generalized Darboux transformation to give the generalized soli-ton formula. Moreover, the breather solutions and rogue wave solutions are presented explicitly. The dynamics of solution is obtained by plotting picture.
第一章为背景介绍.首先介绍方程的背景知识,研究进展.其次介绍反散射方法的历史背景,重要进展和技巧,以及一些最新结果.最后给出了本文的创新点和主要结果.
第二章研究经典Landau-Lifshitz方程的反散射方法.首先运用规范变换和反散射方法适定性理论的新结果,得到Landau-Lifshitz方程在一个带权的Soblev空间的适定性.其次,运用推广的Darboux变换方法将孤立了解给予完全的分类.最后利用Deift-Zhou方法分析一般孤立子解的长时问渐近行为.
第三章研究球对称情形可积的Landau-Lifshitz方程,此模型的可积性研究早在1994年由Lakshamnan等人提出.关于此模型精确解的研究已有些结果,但是利用反散射方法研究该问题尚属首次.主要因为其相应的谱问题为非等谱的半线问题.我们需要对谱问题进行适当的延拓.这里对于奇数维和偶数维系统运用不同的延拓方式.除此之外,我们还对方程解的动力学行为进行分析.同时我们顺便给出了推广的NLS方程的孤子解的动力学行为研究.
第四章研究经典可积Landau-Lifshitz方程在自旋波背景下的反散射问题.运用的主要工具仍然是规范变换和反散射.首先,我们利用规范变换将其变为聚焦的NLS方程在平面波背景下的反散射问题.利用规范变换理论,我们研究其守恒律以及Galilean变换.最后我们利用推广的Darboux变换得到其一般的孤立子解公式.并且具体给出了呼吸子解和怪波解的表达式,同时通过画图分析这些具体解的动力学行为.
In this dissertation, we consider the inverse scattering method of Landau-Lifshitz equation. There are lots of works concerned the integrable Landau-Lifshitz equation in the70s of the last century. Domestic scholars research the relevant issues of this model also in the80s and90s of the last century. Although the classical integrable model is idealization, it provides research idea of the other non-integrable models. Therefore, finding some new phenomenon and new problems in the integrable models can be used to define the research orientation for the non-integrable ones. The main tasks in this dissertation are researching the classical model by inverse scattering method or Riemann-Hilbert approach. On account of the recent developments of inverse scattering method, we can obtain some new results by this approach. Firstly, we can obtain the well-posedness in the frame of inverse scattering. Secondly, we take advantage of the generalized Darboux transformation to tackle with the discrete scattering data and exact solution. We combine generalized Darboux transformation with inverse scattering method for the first time. By this cohesion, we ingeniously deal with the high order pole scattering coefficient. Moreover, the general soliton formula and the complete classification of soliton solution can be obtained. Finally, we can analysis the long time asymptotics of this model by Deift-Zhou method. These results can not obtained by classical PDE analysis. Therefore, the inverse scattering method, as a method to solve the Cauchy problems of integrable partial differential equation, provides the important ways to research it. Recently, there is a hot topic-rogue wave in the nonlinear science. An important physical mechanism to explain the rogue wave is the modulational instability. Owning to the ill-posednees of the model, we can not use the theory of analysis of PDEs in this instance. The inverse scattering method provides new ways to tackle with it.
In chapter1, we briefly introduce the physical background and historical results of the model. Besides, we present the historical background, some new results and important progress with respect to the inverse scattering method. The main innovative points are also presented in this part.
In chapter2, we concern on the inverse scattering method of the classical integrable Landau-Lifshitz equation. Firstly, we obtain the well-posedness in a weighted Soblev space by combining the gauge transformation and some known results. Secondly, we present the completely classification of soliton solution by combining the generalized Darboux transformation and inverse scattering method. Finally, we analyse the long time asymptotics of Landau-Lifshitz equation by Deift-Zhou nonlinear steepest descent method.
In chapter3, we concern on the spherical symmetry Landau-Lifshitz (ssL-L) equa-tion. The integrability of this model is given in by Lakshamnan group. There are some results about exact solution. But there is no result about the Cauchy problem about this model, because it is the non-isospectral half line problem. We need extend the spectral problem on the line properly. Besides, we analyse the dynamics of solution of ssL-L equation. As a by-product, we show the dynamics of the generalized NLS equation.
In chapter4, we consider the inverse scattering problem of Landau-Lifshitz equa-tion on the spin wave background. We give the gauge transformation and inverse scattering analysis. Firstly, we take advantage of gauge transformation to convert this problem into focusing NLS equation on the plane wave background. Besides, we give the Galilean transformation to Landau-Lifshitz equation and conversation laws. Finally, we utilize the generalized Darboux transformation to give the generalized soli-ton formula. Moreover, the breather solutions and rogue wave solutions are presented explicitly. The dynamics of solution is obtained by plotting picture.
引文
[1]陈登远,孤子引论.大学数学丛书,11,科学出版社(2006)
[2]谷超豪,郭柏灵,等等,孤立子理论与应用.浙江科学技术出版社(1990)
[3]郭柏灵,田立新,杨灵娥,殷朝阳,Camassa-Holm方程.现代数学丛书126,科学出版社(2008)
[4]郭柏灵,庞小峰,孤立子.科学出版社(1987)
[5]黄念宁,陈世荣,完全可积非线性方程的哈密顿理论.科学出版社(2005)
[6]M. Ablowitz, D. Kaup, A. Newell and H. Segur, The inverse scattering transform-Fourier analysis for nonlinear problems. Stud. Appl. Math.53,249-315 (1974)
[7]M. J. Ablowitz and H. Segur, Asymptotic Solutions of the Korteweg-deVries Equation. Stud. Appl. Math.57,13-44 (1977)
[8]M. Adler and P. Moerbeke, Birkhoff strata, Baklund transformations, and regulariza-tion of isospectral operators. Adv. Math.108,140-204 (1994)
[9]T. Aktosun, F. Demontis and C. van der Mee, Exact solutions to the focusing nonlinear Schrodinger equation. Inverse Problems 23,2171-2195 (2007)
[10]F. Alouges, A. Soyeur, On global weak solutions for Landau-Lifshitz equations:existence and nonuniquness. Nonlinear Analysis TAM 18,1071-1084 (1992)
[11]R. J. Arms and F. R. Hama, Localized-induction concept on a curved vortex and motion of an elliptic vortex ring. Phys. Fluids 553, (1965)
[12]H. Bart, I. Gohberg, M. Kaashoek, A. Ran, A state space approach to canonical fac-torization with applications. Operator Theory:Advances and Applications Birkhauser (2010)
[13]R. Beals and R.R. Coifman, Scattering and inverse scattering for first order systems. Comm. Pure Appl. Math.37,39-90 (1984)
[14]M. Bertola, A. Tovbis, Universality for the focusing nonlinear Schrodinger equation at the gradient catastrophe point:Rational breathers and poles of the tritronquee solution to Painleve I. Comm. Pure Appl. Math.66,0678-0652 (2013)
[15]G.V. Bezmaternih, Exact solutions of the sine-gordon and landau-lifshitz equations: rational-exponential solutions. Phys. Lett. A 146,492-495 (1990)
[16]D. Bian, B. Guo and L. Ling, Landau-Lishitz equation:High-order soliton and long time asymptotics. Preprint
[17]M. Bogdan and A. Kovalev, Exact multisoliton solution of one-dimensional Landau-Lifshitz equations for an anisotropic ferromagnet. JETP Lett.31,424 (1980)
[18]R. Cascaval, F. Gesztesy, H. Holden and Y. Latushkin, Spectral analysis of Darboux transformations for the focusing NLS hierarchy. J. D'Ana. Math.93,139-197 (2004)
[19]A. Chabchoub, N. Hoffmann, M. Onorato, A. Slunyaev, A. Sergeeva, E. Pelinovsky, and N. Akhmediev, Observation of a hierarchy of up to fifth-order rogue waves in a water tank. Phys. Rev. E 86,056601 (2012)
[20]A. Chabchoub, N. P. Hoffmann, and N. Akhmediev. RogueWave Observation in a Water Wave Tank. Phys. Rev. Lett.106,204502 (2011)
[21]N. Chang, J. Shatah and K. Uhlenbeck, Schroinger Maps. Comm. Pure Appl. Math. 53,0590-0602 (2000)
[22]A. J. Chorin, Equilibrium statistics of a vortex filament with applications. Comm. Math. Phys.141,619-31 (1991)
[23]J. Cielinski, Algebraic construction of the Darboux matrix revisited. J. Phys. A:Math. Theor.42,404003 (2009)
[24]A. Cohen and T. Kappeler, Solutions to the cubic Schodinger equation by the inverse scattering method. SIAM J. Math. Anal.23,900-922 (1992)
[25]G. W. Crabtree and D.R. Nelson, Vortex physics in high-temperature superconductors. Physics Today 50,38-45 (1997)
[26]E. Date, M. Jimbo, M. Kashiwara and T. Miwa, Landau-Lifshitz equation:solitons, quasi-periodic solutions and infinite-dimensional Lie algebras. J. Phys. A:Math. Gen. 16,221 (1983)
[27]M. Daniel, K. Porsezian, and M. Lakshmanan, On the integrability of the inhomoge-neous spherically symmetric Heisenberg ferromagnet in arbitrary dimensions. J. Math. Phys.35,6498 (1994)
[28]P. Deift, Orthogonal polynomials and random matrices:a Riemann-Hilbert approach. Courant Institute of Mathematical Sciences, New York, (1999)
[29]P. Deift and J. Park, Long-time asymptotics for solutions of the NLS equation with a delta potential and even initial data. IMRN 24,5505-5624 (2011)
[30]P. Deift and E. Trubowitz, Inverse Scattering on the Line. Comm. Pure Appl. Math. 32,121-251 (1979)
[31]P. Deift, S. Venakides and X. Zhou, New results in small dispersion KdV by an exten-sion of the steepest descent method for Riemann-Hilbert problems. IMRN 6,285-299 (1997)
[32]P. Deift and X. Zhou, A steepest descent method for oscillatory Riemann Hilbert prob-lems. Asymptotics for the MKdV equation. Anna. Math.137,295-368 (1993)
[33]P. Deift and X. Zhou, Long-time asymptotics for integrable systems. High order theory. Comm. Math. Phys.165,175-191 (1994)
[34]P. Deift and X. Zhou, Perturbation theory for infinite-dimensional integrable systems on the line. A case study. Acta Math.188,163-262 (2002)
[35]P. Deift and X. Zhou, A priori LP estimates for solutions of Riemann-Hilbert problems. IMRN.40,2121-2154 (2002)
[36]P. Deift and X. Zhou, Long-time asymptotics for solutions of the NLS equation with initial data in a weighted Sobolev space. Comm. Pure Appl. Math.56,1029-1077 (2003)
[37]P. Dubard and V.B. Matveev, Multi-rogue waves solutions to the focusing NLS equation. Nat. Hazards Earth Syst. Sci.11,667-672 (2011)
[38]P. Dubard, P. Gaillard, C. Klein and V.B. Matveev, On multi-rogue wave solutions of the NLS equation and positon solutons of the KdV equation. Eur. Phys. J. Special Topics 185,247-258 (2010)
[39]V.G. Dubrovsky and B.G. Konopelchenko, Coherent structures for the Ishimori equa-tion I. Localized solitons with stationary boundaries. Physica D 48,367-395 (1991)
[40]J. J. Duistermaat and F.A. Grunbaum, Differential equations in the spectral parameter. Comm. Math. Phys.103,177-240 (1986)
[41]V.M. Eleonski, I.M. Krichever and N.E. Kulagin, Rational multisoliton solutions of the nonlinear Schrodinger equation. Sov. Phys. Dokl.31,226-228 (1986)
[42]M. Erkintalo, K. Hammani, B. Kibler, C. Finot, N. Akhmediev, J. M. Dudley, and G. Genty, Higher-Order Modulation Instability in Nonlinear Fiber Optics. Phys. Rev. Lett.107,253901 (2011)
[43]L.D. Faddeev and L. A. Takhtajan, Hamiltonian Methods in the Theory of Solitons. Springer-Verlag, Berlin-New York, (1987)
[44]A.S. Fokas, Integrable nonlinear evolution equations on the half-line. Comm. Math. Phys.230,1-39 (2002)
[45]A.S. Fokas, The Davey-Stewartson on the Half-Plane. Comm. Math. Phys.289,957-993 (2009)
[46]A.S. Fokas and A.R. Its, The linearization of the initial-boundary value problem of the nonlinear Schodinger equation. SIAM J. Math. Anal.27,738-764 (1996)
[47]A.P. Fordy and P.P. Kulish, Nonlinear Schrodinger equations and simple Lie algebras. Comm. Math. Phys.89,427-443 (1983)
[48]P. Galliard, Families of quasi-rational solutions of the NLS equation and multi-rogue waves. J.Phys. A:Math. Theor.44,435024 (2011)
[49]L. Gagnon and N. Stievenart, N-soliton interaction in optical fibers:the multiple-pole case. Opt. Lett.11,19 (1986).
[50]Gardner, C. S., Greene, J. M., Kruskal, M. D., and Miura, R. M., Method for solving the Kortcmeg-deVries equation. Phys. Rev. Lett.19,1095-1097 (1967)
[51]F. Gesztesy, W. Karwowski, and Z. Zhao, Limits of soliton solutions. Duke Math. J. 68,101-150 (1992)
[52]I. Gohberg, M.A. Kaashoek and A. Sakhnovich, Pseudo-canonical systems with rational Weyl functions:explicit formulas and applications. J. Diff. Equa.146,375-398 (1998)
[53]K. Grunert and G. Teschl, Long-Time asymptotics for the Kortewegde Vries equation via nonlinear steepest descent. Math. Phys. Anal. Geom.12,287-324 (2009).
[54]C.H. Gu, H.S. Hu, and Z.X. Zhou, Darboux Transformation in Soliton Theory, and its Geometric Applications. Shanghai Science and Technology Publishers, Shanghai (2005)
[55]B. Guo and S. Ding, Landau-Lifshitz equations. Frontiers of Research with the Chinese Academy of Sciences,1, Singapore:World Scientific, (2008)
[56]B. Guo, L. Ling, and Q. P. Liu, Nonlinear Schrodinger equation:Generalized DT and rogue wave solutions. Phys. Rev. E 85,026607 (2012)
[57]B. Guo, L. Ling and Q. P. Liu, High-Order Solutions and Generalized Darboux trans-formations of Derivative Nonlinear Schrodinger Equations. Stud. Appl. Math.130, 317-344 (2013)
[58]B. Guo and L. Ling, Riemann-Hilbert approach and N-soliton formula for coupled derivative Schrodinger equation. J. Math. Phys.53,073506 (2012)
[59]B. Guo and L. Ling, Landau-Lifshitz equation:Riemann-Hilbert method and rogue wave. Preprint.
[60]B. Guo and M. Hong, The Landau-Lifshitz equation of the ferromagnetic spin chain and harmonic maps. Calc. Var.1,311-334 (1993)
[61]S. Gustafson and J. Shatah, The Stability of Localized Solutions of Landau-Lifshitz Equations. Comm. Pure Appl. Math.53,1136-1159 (2002)
[62]R. Hirota, Exact solution of the Kortewegde Vries equation for multiple collisions of solitons. Phys. Rev. Lett.27,1192-1194 (1971)
[63]R. Hirota, Bilinearization of Soliton Equations. J. Phys. Soc. Jpn.51,323-331 (1982)
[64]N. Huang, Z. Chen and Z. Liu, Exact Soliton Solutions for a Spin Chain with an Easy Plane. Phys. Rev. Lett.75,1395-1398 (1995)
[65]S. Kamvissis, Focusing nonlinear Schrodinger equation with infinitely many solitons. J. Math. Phys.36,4175 (1995);
[66]S. Kamvissis, Long Time Behavior for the Focusing Nonlinear Schroedinger Equation with Real Spectral Singularities. Comm. Math. Phys.180,325-341 (1996)
[67]B. Kibler, J. Fatome, C. Finot, G. Millot, F. Dias, G. Genty, N. Akhmediev and J. M. Dudley, The Peregrine soliton in nonlinear fibre optics. Nature Physics 6,790-795 (2010)
[68]H. Kruger and G. Teschl, Long-time asymptotics of the Toda lattice for decaying initial data revisited. Rev. Math. Phys.21,61-109 (2009).
[69]L. Landau and E. Lifshitz, On the theory of the dispersion of magnetic permeability in ferromagnetic bodies. Phys. Zeitsch. der Sow.8,153-169 (1935)
[70]M. Lakshmanan and S. Ganesan, Georetrical and gauge equivalence of the generalized Hirota, Heisenberg and WKIS equations with linear inhomogeneities. Physica 132 A, 117-142 (1985)
[71]M. Lakshmanan, K. Nakanura, Landau-Lifshitz Equation of Ferromagnetism:Exact Treatment of the Gilbert Damping. Phys. Rev. Lett.53,2497-2499 (1984)
[72]M. Lakshmanan and K. Porsezian, Planer radially symmetric Heisenberg spin system and generalized Schrodinger equation:Gauge equivalence, Bdcklund transformations and explicit solutions. Phys. Lett. A 146,329-334 (1990)
[73]M. Lakshmanan, T. Ruijgrok and C. Thompson, On the dynamics of a continuum spin system. Physica 84 A,577-590 (1976)
[74]J. Lee, Analytic properties of zakharov-shabat inverse scattering problem with a poly-nomial. Dissertation, Yale University,1983
[75]W. Liu, W. Zhang, F. Pu and X. Zhou, Nonlinear magnetization dynamics of the classical ferromagnet with two single-ion anisotropies in an external magnetic field. Phys. Rev. B 60,12893-12911 (1999)
[76]W. Liu, B. Wu, X. Zhou, D. Campbell, S. Chui and Q. Niu, Interacting domain walls in an easy-plane ferromagnet. Phys. Rev. B 65,172416 (2002)
[77]V. Matveev and M. Salle, Darboux transformation and solitons. Springer-Verlag, Berlin (1991)
[78]R. Mennichen, A. Sakhnovich and C. Tretter, Direct and inverse spectral problem for a system of differential equations depending rationally on the spectral parameter. Duke Math. J.109,413-449 (2001)
[79]T. Mizumachi and D. Pelinovsky, Backlund Transformation and L2-stability of NLS Solitons. IMRN.9,2034-2067 (2012)
[80]S Novikov, SV Manakov, LP Pitaevskii, VE Zakharov, Theory of solitons:the inverse scattering method. Consultants Bureau, New York (1984)
[81]Tong-ke Ning, Deng-yuan Chen, Da-jun Zhang, The exact solutions for the nonisospec-tral AKNS hierarchy through the inverse scattering transform. Physica A 339,248-266 (2004)
[82]Y. Ohta and J. Yang, Rogue waves in the Davey-Stewartson I equation. Phys. Rev. E. 86,036604 (2012)
[83]Y. Ohta and J. Yang, General high-order rogue wave and their dynamics in the non-linear Schodinger equation. Proc. R. Soc. A 468,1716-1740 (2012)
[84]D. Pelinovsky, Rational solutions of the KP hierarchy and the dynamics of their poles. Ⅱ. Construction of the degenerate polynomial solutions. J. Math. Phys.39,5377-5399 (1998)
[85]K. Porsezian and M. Lakshmanan, On the dynamics of the radially symmetric Heisen-berg ferromagnetic spin system. J. Math. Phys.32,3923-3928 (1991)
[86]R. Radha and M. Lakshmanan, Multisoliton generation in inhomogeneous nonlinear Schrodinger and Heisenberg spin systems. Chaos, Soliton and Fractal 4,181-189 (1994)
[87]R. L. Ricca, Rediscovery of the Da Rios equations. Nature 352,561 (1991)
[88]Y. Rodin, The Riemann boundary problem on Riemann surfaces and the inverse scat-tering problem for the Landau-Lifshitz equation. Physica D 11,90-108 (1984)
[89]A. Sakhnovich, Iterated Bdcklund-Darboux transform for canonical system. J. Func. Anal.144,359-370 (1997)
[90]D. C. Samuels, Vortex filament methods for superfluids Quantized Vortex Dynamics and Superfluid Turbulence. (Lecture Notes in Physics vol 571) (Berlin:Springer) pp 97-113 (2001)
[91]D. H. Sattinger and V.D. Zurkowski, Gauge theory of Bdcklund tranformations. Ⅱ. Physica 27D,225-250 (1987)
[92]P. C. Schuur, Asymptotic analysis of soliton problems. An inverse scattering approach. Lecture Notes in Mathematics,1232. Springer-Verlag, Berlin,1986.
[93]A. Shabat, One dimensional perturbations of a differential operator and the inverse scattering problem. Problems in Mechanics and Mathematical Physics,279-296, Nauka, Moscow, (1976)
[94]L. Sakhnovich, Factorization problems and operator identities. Russ. Math. Surv.41, 1-64 (1986)
[95]H. Segur and M. J. Ablowitz, Asymptotic Solutions and Conservation Laws for the Nonlinear Schrodinger Equation I. J. Math. Phys.17,710-713 (1976)
[96]V.S. Shchesnovich and J. Yang, General soliton matrices in the Riemann-Hilbert prob-lem for integrable nonlinear equations. J. Math. Phys.44,4604 (2003).
[97]V.S. Shchesnovich and J. Yang, Higher-order solitons in the N-wave system. Stud. Appl. Math.110,297 (2003).
[98]Z. S. She, E. Jackson and S. A. Orszag, Intermittent vortex structures in homogeneous isotropic turbulence. Nature 344,226-8 (1990)
[99]E. Sklyanin, On complete integrability of the Landau-Lifshitz equation. LOMI Preprint E-3-79,1979
[100]D. R. Solli, C. Ropers, P. Koonath and B. Jalali, Optical rogue waves. Nature 450, 1054-1057 (2007)
[101]M. Svendsen and H. Fogedby, Phase shift analysis of the Landau-Lifshitz equation. J. Phys. A:Math. Gen.26,1717 (1993)
[102]L.A. Takhtajan, Integration of the continous Heisenberg spin chain through the inverse scattering method. Phys. Lett. A 64 235-237 (1977).
[103]S. Tanaka, Non-linear Schrodinger Equation and Modified Korteweg-de Vries Equation; Construction of Solutions in Terms of Scattering Data. Publ. RIMS, Kyoto Univ.10, 329-357 (1975)
[104]C.-L. Terng and K. Uhlenbeck, Backlund transformations and loop group actions. Comm. Pure Appl. Math.53,1-75 (2000)
[105]A. Tovbis, S. Venakides, X. Zhou, On the long-time limit of semiclassical (zero dis-persion limit) solutions of the focusing nonlinear Schrodinger equation:Pure radiation case. Comm. Pure Appl. Math.57,877-985 (2004)
[106]A. Tovbis, S. Venakides, X. Zhou, On semiclassical (zero dispersion limit) solutions of the focusing nonlinear Schrodinger equation. Comm. Pure Appl. Math.59,1379-1432 (2006)
[107]V. Zakharov and S. Manakov, Asymptotic behavior of nonlinear wave systems integrated by the method of the inverse scattering problem. Sov. Phys. JETP 44,106-112 (1976)
[108]V.E. Zakharov and L.A. Takhtajan, Equivalence of the nonlinear Schodinger equation and the equation of a Heisenberg ferromagnet. Theor. Math. Phys.17-23 (1979)
[109]V. Zakharov and A. Shabat, Exact theory of two-dimensional self-focussing and one-dimensional self-modulating waves in nonlinear media. Sov. Phys.-JETP (Engl. Transl.) 34,62 (1972)
[110]X. Zhou, The Riemann-Hilbert problem and inverse scattering. SIAM J. Math. Anal. 20,966-986 (1989)
[111]X. Zhou, Direct and inverse scattering transforms with arbitrary spectral singularities. Comm. Pure Appl. Math.42,895-938 (1990)
[112]X. Zhou, Inverse scattering for systems with rational spectral dependence. J. Diff. Equa. 115,277-303 (1995)
[113]X. Zhou, L2-Sobolev space bijectivity of the scattering and inverse scattering transforms. Comm. Pure Appl. Math.51,0697-0731 (1998)
[114]Y. Zhou, B. Guo and S. Tan, Existence and uniqueness of smooth solution for system of ferromagnetic chain. Seience in China A 34,257-266 (1991)
[115]J.P. Zubelli, Rational solutions of nonlinear evolution equations, vertex operators, and bispectrality. J. Diff. Equa.97,71-98 (1992)
[116]J.P. Zubelli and F. Magri, Differential equations in the spectral paramters, Darboux transformations and a hierarchy of master symmetries for KdV. Comm. Math. Phys. 141,329-351 (1991)
[2]谷超豪,郭柏灵,等等,孤立子理论与应用.浙江科学技术出版社(1990)
[3]郭柏灵,田立新,杨灵娥,殷朝阳,Camassa-Holm方程.现代数学丛书126,科学出版社(2008)
[4]郭柏灵,庞小峰,孤立子.科学出版社(1987)
[5]黄念宁,陈世荣,完全可积非线性方程的哈密顿理论.科学出版社(2005)
[6]M. Ablowitz, D. Kaup, A. Newell and H. Segur, The inverse scattering transform-Fourier analysis for nonlinear problems. Stud. Appl. Math.53,249-315 (1974)
[7]M. J. Ablowitz and H. Segur, Asymptotic Solutions of the Korteweg-deVries Equation. Stud. Appl. Math.57,13-44 (1977)
[8]M. Adler and P. Moerbeke, Birkhoff strata, Baklund transformations, and regulariza-tion of isospectral operators. Adv. Math.108,140-204 (1994)
[9]T. Aktosun, F. Demontis and C. van der Mee, Exact solutions to the focusing nonlinear Schrodinger equation. Inverse Problems 23,2171-2195 (2007)
[10]F. Alouges, A. Soyeur, On global weak solutions for Landau-Lifshitz equations:existence and nonuniquness. Nonlinear Analysis TAM 18,1071-1084 (1992)
[11]R. J. Arms and F. R. Hama, Localized-induction concept on a curved vortex and motion of an elliptic vortex ring. Phys. Fluids 553, (1965)
[12]H. Bart, I. Gohberg, M. Kaashoek, A. Ran, A state space approach to canonical fac-torization with applications. Operator Theory:Advances and Applications Birkhauser (2010)
[13]R. Beals and R.R. Coifman, Scattering and inverse scattering for first order systems. Comm. Pure Appl. Math.37,39-90 (1984)
[14]M. Bertola, A. Tovbis, Universality for the focusing nonlinear Schrodinger equation at the gradient catastrophe point:Rational breathers and poles of the tritronquee solution to Painleve I. Comm. Pure Appl. Math.66,0678-0652 (2013)
[15]G.V. Bezmaternih, Exact solutions of the sine-gordon and landau-lifshitz equations: rational-exponential solutions. Phys. Lett. A 146,492-495 (1990)
[16]D. Bian, B. Guo and L. Ling, Landau-Lishitz equation:High-order soliton and long time asymptotics. Preprint
[17]M. Bogdan and A. Kovalev, Exact multisoliton solution of one-dimensional Landau-Lifshitz equations for an anisotropic ferromagnet. JETP Lett.31,424 (1980)
[18]R. Cascaval, F. Gesztesy, H. Holden and Y. Latushkin, Spectral analysis of Darboux transformations for the focusing NLS hierarchy. J. D'Ana. Math.93,139-197 (2004)
[19]A. Chabchoub, N. Hoffmann, M. Onorato, A. Slunyaev, A. Sergeeva, E. Pelinovsky, and N. Akhmediev, Observation of a hierarchy of up to fifth-order rogue waves in a water tank. Phys. Rev. E 86,056601 (2012)
[20]A. Chabchoub, N. P. Hoffmann, and N. Akhmediev. RogueWave Observation in a Water Wave Tank. Phys. Rev. Lett.106,204502 (2011)
[21]N. Chang, J. Shatah and K. Uhlenbeck, Schroinger Maps. Comm. Pure Appl. Math. 53,0590-0602 (2000)
[22]A. J. Chorin, Equilibrium statistics of a vortex filament with applications. Comm. Math. Phys.141,619-31 (1991)
[23]J. Cielinski, Algebraic construction of the Darboux matrix revisited. J. Phys. A:Math. Theor.42,404003 (2009)
[24]A. Cohen and T. Kappeler, Solutions to the cubic Schodinger equation by the inverse scattering method. SIAM J. Math. Anal.23,900-922 (1992)
[25]G. W. Crabtree and D.R. Nelson, Vortex physics in high-temperature superconductors. Physics Today 50,38-45 (1997)
[26]E. Date, M. Jimbo, M. Kashiwara and T. Miwa, Landau-Lifshitz equation:solitons, quasi-periodic solutions and infinite-dimensional Lie algebras. J. Phys. A:Math. Gen. 16,221 (1983)
[27]M. Daniel, K. Porsezian, and M. Lakshmanan, On the integrability of the inhomoge-neous spherically symmetric Heisenberg ferromagnet in arbitrary dimensions. J. Math. Phys.35,6498 (1994)
[28]P. Deift, Orthogonal polynomials and random matrices:a Riemann-Hilbert approach. Courant Institute of Mathematical Sciences, New York, (1999)
[29]P. Deift and J. Park, Long-time asymptotics for solutions of the NLS equation with a delta potential and even initial data. IMRN 24,5505-5624 (2011)
[30]P. Deift and E. Trubowitz, Inverse Scattering on the Line. Comm. Pure Appl. Math. 32,121-251 (1979)
[31]P. Deift, S. Venakides and X. Zhou, New results in small dispersion KdV by an exten-sion of the steepest descent method for Riemann-Hilbert problems. IMRN 6,285-299 (1997)
[32]P. Deift and X. Zhou, A steepest descent method for oscillatory Riemann Hilbert prob-lems. Asymptotics for the MKdV equation. Anna. Math.137,295-368 (1993)
[33]P. Deift and X. Zhou, Long-time asymptotics for integrable systems. High order theory. Comm. Math. Phys.165,175-191 (1994)
[34]P. Deift and X. Zhou, Perturbation theory for infinite-dimensional integrable systems on the line. A case study. Acta Math.188,163-262 (2002)
[35]P. Deift and X. Zhou, A priori LP estimates for solutions of Riemann-Hilbert problems. IMRN.40,2121-2154 (2002)
[36]P. Deift and X. Zhou, Long-time asymptotics for solutions of the NLS equation with initial data in a weighted Sobolev space. Comm. Pure Appl. Math.56,1029-1077 (2003)
[37]P. Dubard and V.B. Matveev, Multi-rogue waves solutions to the focusing NLS equation. Nat. Hazards Earth Syst. Sci.11,667-672 (2011)
[38]P. Dubard, P. Gaillard, C. Klein and V.B. Matveev, On multi-rogue wave solutions of the NLS equation and positon solutons of the KdV equation. Eur. Phys. J. Special Topics 185,247-258 (2010)
[39]V.G. Dubrovsky and B.G. Konopelchenko, Coherent structures for the Ishimori equa-tion I. Localized solitons with stationary boundaries. Physica D 48,367-395 (1991)
[40]J. J. Duistermaat and F.A. Grunbaum, Differential equations in the spectral parameter. Comm. Math. Phys.103,177-240 (1986)
[41]V.M. Eleonski, I.M. Krichever and N.E. Kulagin, Rational multisoliton solutions of the nonlinear Schrodinger equation. Sov. Phys. Dokl.31,226-228 (1986)
[42]M. Erkintalo, K. Hammani, B. Kibler, C. Finot, N. Akhmediev, J. M. Dudley, and G. Genty, Higher-Order Modulation Instability in Nonlinear Fiber Optics. Phys. Rev. Lett.107,253901 (2011)
[43]L.D. Faddeev and L. A. Takhtajan, Hamiltonian Methods in the Theory of Solitons. Springer-Verlag, Berlin-New York, (1987)
[44]A.S. Fokas, Integrable nonlinear evolution equations on the half-line. Comm. Math. Phys.230,1-39 (2002)
[45]A.S. Fokas, The Davey-Stewartson on the Half-Plane. Comm. Math. Phys.289,957-993 (2009)
[46]A.S. Fokas and A.R. Its, The linearization of the initial-boundary value problem of the nonlinear Schodinger equation. SIAM J. Math. Anal.27,738-764 (1996)
[47]A.P. Fordy and P.P. Kulish, Nonlinear Schrodinger equations and simple Lie algebras. Comm. Math. Phys.89,427-443 (1983)
[48]P. Galliard, Families of quasi-rational solutions of the NLS equation and multi-rogue waves. J.Phys. A:Math. Theor.44,435024 (2011)
[49]L. Gagnon and N. Stievenart, N-soliton interaction in optical fibers:the multiple-pole case. Opt. Lett.11,19 (1986).
[50]Gardner, C. S., Greene, J. M., Kruskal, M. D., and Miura, R. M., Method for solving the Kortcmeg-deVries equation. Phys. Rev. Lett.19,1095-1097 (1967)
[51]F. Gesztesy, W. Karwowski, and Z. Zhao, Limits of soliton solutions. Duke Math. J. 68,101-150 (1992)
[52]I. Gohberg, M.A. Kaashoek and A. Sakhnovich, Pseudo-canonical systems with rational Weyl functions:explicit formulas and applications. J. Diff. Equa.146,375-398 (1998)
[53]K. Grunert and G. Teschl, Long-Time asymptotics for the Kortewegde Vries equation via nonlinear steepest descent. Math. Phys. Anal. Geom.12,287-324 (2009).
[54]C.H. Gu, H.S. Hu, and Z.X. Zhou, Darboux Transformation in Soliton Theory, and its Geometric Applications. Shanghai Science and Technology Publishers, Shanghai (2005)
[55]B. Guo and S. Ding, Landau-Lifshitz equations. Frontiers of Research with the Chinese Academy of Sciences,1, Singapore:World Scientific, (2008)
[56]B. Guo, L. Ling, and Q. P. Liu, Nonlinear Schrodinger equation:Generalized DT and rogue wave solutions. Phys. Rev. E 85,026607 (2012)
[57]B. Guo, L. Ling and Q. P. Liu, High-Order Solutions and Generalized Darboux trans-formations of Derivative Nonlinear Schrodinger Equations. Stud. Appl. Math.130, 317-344 (2013)
[58]B. Guo and L. Ling, Riemann-Hilbert approach and N-soliton formula for coupled derivative Schrodinger equation. J. Math. Phys.53,073506 (2012)
[59]B. Guo and L. Ling, Landau-Lifshitz equation:Riemann-Hilbert method and rogue wave. Preprint.
[60]B. Guo and M. Hong, The Landau-Lifshitz equation of the ferromagnetic spin chain and harmonic maps. Calc. Var.1,311-334 (1993)
[61]S. Gustafson and J. Shatah, The Stability of Localized Solutions of Landau-Lifshitz Equations. Comm. Pure Appl. Math.53,1136-1159 (2002)
[62]R. Hirota, Exact solution of the Kortewegde Vries equation for multiple collisions of solitons. Phys. Rev. Lett.27,1192-1194 (1971)
[63]R. Hirota, Bilinearization of Soliton Equations. J. Phys. Soc. Jpn.51,323-331 (1982)
[64]N. Huang, Z. Chen and Z. Liu, Exact Soliton Solutions for a Spin Chain with an Easy Plane. Phys. Rev. Lett.75,1395-1398 (1995)
[65]S. Kamvissis, Focusing nonlinear Schrodinger equation with infinitely many solitons. J. Math. Phys.36,4175 (1995);
[66]S. Kamvissis, Long Time Behavior for the Focusing Nonlinear Schroedinger Equation with Real Spectral Singularities. Comm. Math. Phys.180,325-341 (1996)
[67]B. Kibler, J. Fatome, C. Finot, G. Millot, F. Dias, G. Genty, N. Akhmediev and J. M. Dudley, The Peregrine soliton in nonlinear fibre optics. Nature Physics 6,790-795 (2010)
[68]H. Kruger and G. Teschl, Long-time asymptotics of the Toda lattice for decaying initial data revisited. Rev. Math. Phys.21,61-109 (2009).
[69]L. Landau and E. Lifshitz, On the theory of the dispersion of magnetic permeability in ferromagnetic bodies. Phys. Zeitsch. der Sow.8,153-169 (1935)
[70]M. Lakshmanan and S. Ganesan, Georetrical and gauge equivalence of the generalized Hirota, Heisenberg and WKIS equations with linear inhomogeneities. Physica 132 A, 117-142 (1985)
[71]M. Lakshmanan, K. Nakanura, Landau-Lifshitz Equation of Ferromagnetism:Exact Treatment of the Gilbert Damping. Phys. Rev. Lett.53,2497-2499 (1984)
[72]M. Lakshmanan and K. Porsezian, Planer radially symmetric Heisenberg spin system and generalized Schrodinger equation:Gauge equivalence, Bdcklund transformations and explicit solutions. Phys. Lett. A 146,329-334 (1990)
[73]M. Lakshmanan, T. Ruijgrok and C. Thompson, On the dynamics of a continuum spin system. Physica 84 A,577-590 (1976)
[74]J. Lee, Analytic properties of zakharov-shabat inverse scattering problem with a poly-nomial. Dissertation, Yale University,1983
[75]W. Liu, W. Zhang, F. Pu and X. Zhou, Nonlinear magnetization dynamics of the classical ferromagnet with two single-ion anisotropies in an external magnetic field. Phys. Rev. B 60,12893-12911 (1999)
[76]W. Liu, B. Wu, X. Zhou, D. Campbell, S. Chui and Q. Niu, Interacting domain walls in an easy-plane ferromagnet. Phys. Rev. B 65,172416 (2002)
[77]V. Matveev and M. Salle, Darboux transformation and solitons. Springer-Verlag, Berlin (1991)
[78]R. Mennichen, A. Sakhnovich and C. Tretter, Direct and inverse spectral problem for a system of differential equations depending rationally on the spectral parameter. Duke Math. J.109,413-449 (2001)
[79]T. Mizumachi and D. Pelinovsky, Backlund Transformation and L2-stability of NLS Solitons. IMRN.9,2034-2067 (2012)
[80]S Novikov, SV Manakov, LP Pitaevskii, VE Zakharov, Theory of solitons:the inverse scattering method. Consultants Bureau, New York (1984)
[81]Tong-ke Ning, Deng-yuan Chen, Da-jun Zhang, The exact solutions for the nonisospec-tral AKNS hierarchy through the inverse scattering transform. Physica A 339,248-266 (2004)
[82]Y. Ohta and J. Yang, Rogue waves in the Davey-Stewartson I equation. Phys. Rev. E. 86,036604 (2012)
[83]Y. Ohta and J. Yang, General high-order rogue wave and their dynamics in the non-linear Schodinger equation. Proc. R. Soc. A 468,1716-1740 (2012)
[84]D. Pelinovsky, Rational solutions of the KP hierarchy and the dynamics of their poles. Ⅱ. Construction of the degenerate polynomial solutions. J. Math. Phys.39,5377-5399 (1998)
[85]K. Porsezian and M. Lakshmanan, On the dynamics of the radially symmetric Heisen-berg ferromagnetic spin system. J. Math. Phys.32,3923-3928 (1991)
[86]R. Radha and M. Lakshmanan, Multisoliton generation in inhomogeneous nonlinear Schrodinger and Heisenberg spin systems. Chaos, Soliton and Fractal 4,181-189 (1994)
[87]R. L. Ricca, Rediscovery of the Da Rios equations. Nature 352,561 (1991)
[88]Y. Rodin, The Riemann boundary problem on Riemann surfaces and the inverse scat-tering problem for the Landau-Lifshitz equation. Physica D 11,90-108 (1984)
[89]A. Sakhnovich, Iterated Bdcklund-Darboux transform for canonical system. J. Func. Anal.144,359-370 (1997)
[90]D. C. Samuels, Vortex filament methods for superfluids Quantized Vortex Dynamics and Superfluid Turbulence. (Lecture Notes in Physics vol 571) (Berlin:Springer) pp 97-113 (2001)
[91]D. H. Sattinger and V.D. Zurkowski, Gauge theory of Bdcklund tranformations. Ⅱ. Physica 27D,225-250 (1987)
[92]P. C. Schuur, Asymptotic analysis of soliton problems. An inverse scattering approach. Lecture Notes in Mathematics,1232. Springer-Verlag, Berlin,1986.
[93]A. Shabat, One dimensional perturbations of a differential operator and the inverse scattering problem. Problems in Mechanics and Mathematical Physics,279-296, Nauka, Moscow, (1976)
[94]L. Sakhnovich, Factorization problems and operator identities. Russ. Math. Surv.41, 1-64 (1986)
[95]H. Segur and M. J. Ablowitz, Asymptotic Solutions and Conservation Laws for the Nonlinear Schrodinger Equation I. J. Math. Phys.17,710-713 (1976)
[96]V.S. Shchesnovich and J. Yang, General soliton matrices in the Riemann-Hilbert prob-lem for integrable nonlinear equations. J. Math. Phys.44,4604 (2003).
[97]V.S. Shchesnovich and J. Yang, Higher-order solitons in the N-wave system. Stud. Appl. Math.110,297 (2003).
[98]Z. S. She, E. Jackson and S. A. Orszag, Intermittent vortex structures in homogeneous isotropic turbulence. Nature 344,226-8 (1990)
[99]E. Sklyanin, On complete integrability of the Landau-Lifshitz equation. LOMI Preprint E-3-79,1979
[100]D. R. Solli, C. Ropers, P. Koonath and B. Jalali, Optical rogue waves. Nature 450, 1054-1057 (2007)
[101]M. Svendsen and H. Fogedby, Phase shift analysis of the Landau-Lifshitz equation. J. Phys. A:Math. Gen.26,1717 (1993)
[102]L.A. Takhtajan, Integration of the continous Heisenberg spin chain through the inverse scattering method. Phys. Lett. A 64 235-237 (1977).
[103]S. Tanaka, Non-linear Schrodinger Equation and Modified Korteweg-de Vries Equation; Construction of Solutions in Terms of Scattering Data. Publ. RIMS, Kyoto Univ.10, 329-357 (1975)
[104]C.-L. Terng and K. Uhlenbeck, Backlund transformations and loop group actions. Comm. Pure Appl. Math.53,1-75 (2000)
[105]A. Tovbis, S. Venakides, X. Zhou, On the long-time limit of semiclassical (zero dis-persion limit) solutions of the focusing nonlinear Schrodinger equation:Pure radiation case. Comm. Pure Appl. Math.57,877-985 (2004)
[106]A. Tovbis, S. Venakides, X. Zhou, On semiclassical (zero dispersion limit) solutions of the focusing nonlinear Schrodinger equation. Comm. Pure Appl. Math.59,1379-1432 (2006)
[107]V. Zakharov and S. Manakov, Asymptotic behavior of nonlinear wave systems integrated by the method of the inverse scattering problem. Sov. Phys. JETP 44,106-112 (1976)
[108]V.E. Zakharov and L.A. Takhtajan, Equivalence of the nonlinear Schodinger equation and the equation of a Heisenberg ferromagnet. Theor. Math. Phys.17-23 (1979)
[109]V. Zakharov and A. Shabat, Exact theory of two-dimensional self-focussing and one-dimensional self-modulating waves in nonlinear media. Sov. Phys.-JETP (Engl. Transl.) 34,62 (1972)
[110]X. Zhou, The Riemann-Hilbert problem and inverse scattering. SIAM J. Math. Anal. 20,966-986 (1989)
[111]X. Zhou, Direct and inverse scattering transforms with arbitrary spectral singularities. Comm. Pure Appl. Math.42,895-938 (1990)
[112]X. Zhou, Inverse scattering for systems with rational spectral dependence. J. Diff. Equa. 115,277-303 (1995)
[113]X. Zhou, L2-Sobolev space bijectivity of the scattering and inverse scattering transforms. Comm. Pure Appl. Math.51,0697-0731 (1998)
[114]Y. Zhou, B. Guo and S. Tan, Existence and uniqueness of smooth solution for system of ferromagnetic chain. Seience in China A 34,257-266 (1991)
[115]J.P. Zubelli, Rational solutions of nonlinear evolution equations, vertex operators, and bispectrality. J. Diff. Equa.97,71-98 (1992)
[116]J.P. Zubelli and F. Magri, Differential equations in the spectral paramters, Darboux transformations and a hierarchy of master symmetries for KdV. Comm. Math. Phys. 141,329-351 (1991)