一类非线性浅水波方程的整体守恒解及非连续解理论
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摘要
本文研究了广义Camassa-Holm方程、Degasperis-Procesi方程的整体守恒解,以及新型双Sine-Gordon方程的不连续解。Camassa和Holm利用哈密顿方法获得了一类新型色散波方程,叫Camassa-Holm方程(简称CH方程),它具有双哈密顿结构和无穷多守恒量,是完全可积的。Degasperis-Procesi方程(简称DP方程)是Degasperis和Procesi得到的,它不仅有尖峰解,还有激波解。双Sine-Gordon方程是一个很重要的方程,因为它广泛应用于诸如非线性光学等领域。最近,双Sine-Gordon方程的一些精确解已经得到。
第三章主要研究广义CH方程初值问题的整体守恒解,先将这个方程转化成一个常微分系统。在这个常微分系统中,应用索伯列夫空间的一些不等式、常微分方程相关知识,讨论解的适定性问题;
第四章主要研究了DP方程初值问题的适定性问题,采用了一个不同于CH方程的守恒律来辅助证明,证明了短时期解的存在性,进而证明了它的整体守恒解存在且唯一;
第五章主要研究了新型双Sine-Gordon方程的精确解,我们发现有两个精确解是不连续的,进而通过守恒律方程理论证明了这两个解是非连续的。
In this paper,we study the existence of global conservative solutions of the Cauchy problem for the generalized Camassa-Holm equation and Degasperis-Procesesi equation,and the exact solutions of the new double Sine-Gordon equation.Camassa and Holm derived a new completely integrable dispersive wave equation for water waves by Hamiltonian method,namely Camassa-Holm equation(i.e.,CH equation).It has bi-hamiltonian structure and infinite sequence of conserved quantities, and it is completely integrable.Degasperis-Procesi equation(i.e.,DP equation) was found by Degasperis and Procesi.It has not only peakon wave solutions,but also shock wave solutions.The double Sine-Gordon equation is a significant equation because of its applications in many fields such as nonlinear optics.Recently,many exact solutions of the double Sine-Gordon equation were found.
In Chapter Three,we prove the existence of global conservative solutions of the Cauchy problem for the generalized Camassa-Holm equation.We change it into an ODE system in a Banach space.In this system,by applying some inequalities in Sobolev space and some knowledge about the ODE theories,we consider the well-posedness of the solutions.
In Chapter Four,we study the well-posedness of the DP equation.We choose a conservative law which is different from CH equation to finish our proof.Then we prove the existence and uniqueness of the short-time solutions.Hence the existence of global conservative solutions with respect to the initial date is obtained.
In Chapter Five,we study the exact solutions for new double Sine-Gordon equation,we find that two exact solutions of the equation are noncontinous.Then we prove that the two solutions are noncontinuous solutions by conservation equation theory.
第三章主要研究广义CH方程初值问题的整体守恒解,先将这个方程转化成一个常微分系统。在这个常微分系统中,应用索伯列夫空间的一些不等式、常微分方程相关知识,讨论解的适定性问题;
第四章主要研究了DP方程初值问题的适定性问题,采用了一个不同于CH方程的守恒律来辅助证明,证明了短时期解的存在性,进而证明了它的整体守恒解存在且唯一;
第五章主要研究了新型双Sine-Gordon方程的精确解,我们发现有两个精确解是不连续的,进而通过守恒律方程理论证明了这两个解是非连续的。
In this paper,we study the existence of global conservative solutions of the Cauchy problem for the generalized Camassa-Holm equation and Degasperis-Procesesi equation,and the exact solutions of the new double Sine-Gordon equation.Camassa and Holm derived a new completely integrable dispersive wave equation for water waves by Hamiltonian method,namely Camassa-Holm equation(i.e.,CH equation).It has bi-hamiltonian structure and infinite sequence of conserved quantities, and it is completely integrable.Degasperis-Procesi equation(i.e.,DP equation) was found by Degasperis and Procesi.It has not only peakon wave solutions,but also shock wave solutions.The double Sine-Gordon equation is a significant equation because of its applications in many fields such as nonlinear optics.Recently,many exact solutions of the double Sine-Gordon equation were found.
In Chapter Three,we prove the existence of global conservative solutions of the Cauchy problem for the generalized Camassa-Holm equation.We change it into an ODE system in a Banach space.In this system,by applying some inequalities in Sobolev space and some knowledge about the ODE theories,we consider the well-posedness of the solutions.
In Chapter Four,we study the well-posedness of the DP equation.We choose a conservative law which is different from CH equation to finish our proof.Then we prove the existence and uniqueness of the short-time solutions.Hence the existence of global conservative solutions with respect to the initial date is obtained.
In Chapter Five,we study the exact solutions for new double Sine-Gordon equation,we find that two exact solutions of the equation are noncontinous.Then we prove that the two solutions are noncontinuous solutions by conservation equation theory.
引文
[1]Camassa R.,Holm D.D:An integrable shallow water equation with peaked solitons.Phys.Rev.Lett.1993,71(11),1661-1664.
[2]Fuchssteiner B.,Fokas A.S.:Symplectic structures,their Backlund transformation andheredi- tary symmetries.Physica D.1981,4,47-66.
[3]Johnson R.S.,Camassa-Holm:Korteweg-de Vries and related models for water waves,J.Fluid.Mech.2002,457,63-82.
[4]Dullin R.,Gottwald G and Holm D:An integrable shallow water equation with linear andnonlinear dispersion.Phys.Rev.Lett.2001,87(9),4501-4504.
[5]Camassa,R.,Holm D.,Hyman,J.:A new integrable shallow water equation.Adv.Appl.Mech.,1994,31,1-33.
[6]Fisher M.,Schiff J.:The Camassa Holm equations:conserved quantities and the initialvalue problem.Phy.Lett.A.1999,259(3),371-376.
[7]Clarkson P.A.,Mansfield EX.,Priestley T.J.:Symmetries of a class of nonlinear third-orderpartial differential equations.Math.Comput.Modelling.1997,25 (8-9),195-212.
[8]F.Cooper,H.Shepard:Solitons in the Camassa-Holm shallow water equation.Physics.Letters.A.1994,194(4),246-250.
[9]Lixin Tian,Gang Xu,Zengrong Liu:The concave or convex peaked and smooth solitonsolu-tions of Camassa-Holm equation,Applied Math.Mech.2002,123(5),557-567.
[10]Lixin Tian,Xiuying Song:New peaked solitary wave solutions of the generalized Camassa-Holm equation.Chaos,Solitons and Fractals.2004,19(3),621-637.
[11]Lixin Tian,Jiuli Yin:New compacton solutions and solitary solutions of fully nonlinear generalized Camassa-Holm equations.Chaos,Soliton and Fractals.2004,20(4),289-299.
[12]Constantin A and Escher J:Global existence and blow-up for a shallow water equation.Ann.Sc.Norm.Sup.Pisa.1998,26,303-328.
[13]Constantin A:Global existence of solutions and breaking waves for a shallow waterequation:a geometric approach.AnnJnst.Fourier (Grenoble).2000,50,321-362.
[14]Constantin A:The Hamitonian structure of the Camassa-Holm equation,Exposition.Math.1997,15,53-85.
[15]Constantin A,H.P.McKean:A shallow water equation on the circle,Comm.Pure Appl.Math.1999,52(8),949-982.
[16]Constantin A.,Strauss,W.A.:Stability of the Camassa-Holm solitons.J.Nonlinear Sci.2002,12,415-422.
[17]Constantin A:On the scattering problem for the Camassa-Holm equation.Proc.R.Soc.London A.2001,457,953-970.
[18]Lenells J:The Scattering approach for the Camassa-Holm equation.J.Nonliear Math.Phys. 2002,9(4),389-393.
[19]Beals R.,Sattinger D and Szmigielski J:Acoustic scattering and the extended KortewegdeVries hierarchy.Adv.Math.,1998,140,190-206.
[20]Constantin A and Escher J:Wave Breaking for Nonlinear Nonlocal Shallow Water Equations,Acta Mathematica.1998,181,229-243.
[21]Danchin R:A Few Remarks on the Camassa-Holm Equation,Differential and Integral Equations.2001,14,953-988.
[22]Guo BL,Liu ZR.:Peaked wave solutions of CH-r equation,Sci China (Ser..A).2003,33(4),325-337.
[23]Minying Tang,Chengxi Yang:Extension on peaked wave solutions of CH-r equation.Chaos,Solitons and Fractals.2004,20,815-825.
[24]J.Bona,R.Smith:The initial-value problem for the Korteweg-de Vries equation.Philos.Trans.Royal Soc.London Series A.1975,278,555-601.
[25]A.Degasperis,M.Proceci:Asymptotic integrability,in Symmetry and Perturbation Theory,edited by A.Degasperis and G.Gaeta,World Scientific(1999),23-37.
[26]P.G.Drazin and R.SJohnson,Solitions:an Introduction,Cambridge University press,Cambridge-New York,1989.
[27]H.P.Mckean:Integrable systems and algebraic curves.[J].GlobalAnalysis.Springer Lecture Notes in Mathematicas.1979,755,83-200.
[28]C.Kenig,G.Ponce,and L.Vega:Well-posedness and scattering results for the generalized Korteweg-de Veris equation via the contraction principle.[J].Comm.pure Appl.Math.1993,46,527-620.
[29]A.Constantin and J.Escher:Global existence and blow-up for a shallow water equation.[J].Annali Sc.Norm.Sup.Pisa.1998,26,303-328.
[30]Y.Li and P.Olver:Well-posedness and blow-up solutions for an integrable nonlinearly dispersive model wave equation.[J].J.Differential Equations.2000,162,27-63.
[31]A.Constantin:Global existence of solutions and breaking waves for a shallow water equations geometric approach.[J].Ann.Inst.Fourier(Grenoble).2000,50,321-362.
[32]A.Constantin and H.P.McKean:A shallow water equation on the circle.[J].Comm,Pure Appl.Math.1999,52,949-982.
[33]A.Degasperis,D.D.Holm,A.N.W.Hone:A New Integral Equation with Peakon Solutions[J].Theoretical and Mathematical Physics,2002,133,1463-1474.
[34]H.R.Dullin,G.A.Gottwald and D.D.Holm:Camassa-Holm,Korteweg-de Veris-5 and other asymptotically equivalent equations for shallow water waves.[J].Fluid Dynamics Research.2003,33,73-79.
[35]V.O.Vakhnenko,EJ.Parkes:Periodic and solitary-wave solutions of the Degasperis- Procesi equation.[JJ.Chaos Solitons Fractals.2004,20,1059-1073.
[36]Z.Yin:On the Cauchy problem for an integrable equation with peakon solutions.[J].Illinois J.Math.2003,47(1),649-666.
[37]Z.Yin:Global weak solutions to a new periodic integrable equation with peakon solutions.[J].J.Funct.Anal.2004,212,182-194.
[38]Z.Yin:Global solutions to a new integrable equation with peakons.[J].Indiana Univ.Math.J.2004,53,1189-1210.
[39]J.Lenells:Traveling wave solutions of the Degasperis-Procesi equation.[J].J.Math.Anal.Appl.2005,306,72-82.
[40]Y.Matsuno:Multisolition solutions of the Degasperis-Procesi equation and their peakon limit.[J].Invere Problems,2005,21,1553-1570.
[41]Constantine A.Popov:Perturbation theory for the doule Sine-Gordon equation.Wave Motion.2005,Vol 42,4,309-316.
[42]Mingling Wang,Xiangzheng Li:Exact solutions to the double Sine-Gordon equation.Chaos,Solitons and Fractals.2006,Vol 27,Issue 2,477-486.
[43]Shuimeng Yu,Weihua Yang,Jinxian Ruan,Lixin Tian:Solitary Wave Solutions to Approximate Fully Nonlinear Double Sine-Gordon Equation.International Journal of Nonlinear Science.2007,Vol 3,No3,163-169.
[44]Rosenau P,Hyman J.M.:Compactons.solitons with finite wavelength.[J],Phys.Rev.Lett.1993,70,564-567.
[45]Lixin Tian,Jiuli Yin:New peakon and multi-compacton solitary wave solutions of fully nonlinear Sine-Gordon equation.Chaos,Solitons and Fractals.2005,Vol 24,Issuel,353-363.
[46]Jiuli Yin,Lixin Tian:Peakon and Compacton Solutions for K (p q,\) Equation.International Journal of Nonlinear Science.2007,Vol 3,No.2,133-136.
[47]Lixin Tian,Xiuying Song:New peaked solitary wave solutions of the generalized Camassa-Holm equation.Chaos,Solitons and Fractals.2004,Vol 19,Issue 3,621 -637.
[48]Shuyi Zhang:Discontinous solutions to the Euler equations for a van der waals fluid.Appl.Math.Lett.2007,20,170-176.
[49]Lixin Tian,Shuimeng Yu,Meijie Ni:Noncontinuous Solitary Wave solutions for Double Sine-Gordon Equation.Chinese Physics Letters.2007,Vol.24,No.7,1791-1794.
[50]G.M.Coclite,H.Holden,K.H.Karlsen:Well-posedness for a parabolic-elliptic system,Discrete Contin.Dyn.Syst.2005,13,659-682.
[51]G.M.Coclite,H.Holden,K.H.Karlsen:Global weak solutions to a generalized hyperelasticrod wave equation,SIAM J.Math.Anal.2005,37,1044-1069.
[52]M.Fonte:Conservative solution of the Camassa-Holm equation on the real line,math.AP/0511549.
[53]H.Holden,X.Raynaud:Global conservative solutions of the Camassa-Holm equation-a Lagrangian point of view,Comm.Partial Differential Equations,in press.
[54]G.B.Folland:Real Analysis,seconded.,Wiley,New York,1999.
[55]H.Holden,X.Raynaud:Global conservative solutions of the generalized hyperelastic-rod wave equation,J.Differential Equations.2007,233,448-484.
[2]Fuchssteiner B.,Fokas A.S.:Symplectic structures,their Backlund transformation andheredi- tary symmetries.Physica D.1981,4,47-66.
[3]Johnson R.S.,Camassa-Holm:Korteweg-de Vries and related models for water waves,J.Fluid.Mech.2002,457,63-82.
[4]Dullin R.,Gottwald G and Holm D:An integrable shallow water equation with linear andnonlinear dispersion.Phys.Rev.Lett.2001,87(9),4501-4504.
[5]Camassa,R.,Holm D.,Hyman,J.:A new integrable shallow water equation.Adv.Appl.Mech.,1994,31,1-33.
[6]Fisher M.,Schiff J.:The Camassa Holm equations:conserved quantities and the initialvalue problem.Phy.Lett.A.1999,259(3),371-376.
[7]Clarkson P.A.,Mansfield EX.,Priestley T.J.:Symmetries of a class of nonlinear third-orderpartial differential equations.Math.Comput.Modelling.1997,25 (8-9),195-212.
[8]F.Cooper,H.Shepard:Solitons in the Camassa-Holm shallow water equation.Physics.Letters.A.1994,194(4),246-250.
[9]Lixin Tian,Gang Xu,Zengrong Liu:The concave or convex peaked and smooth solitonsolu-tions of Camassa-Holm equation,Applied Math.Mech.2002,123(5),557-567.
[10]Lixin Tian,Xiuying Song:New peaked solitary wave solutions of the generalized Camassa-Holm equation.Chaos,Solitons and Fractals.2004,19(3),621-637.
[11]Lixin Tian,Jiuli Yin:New compacton solutions and solitary solutions of fully nonlinear generalized Camassa-Holm equations.Chaos,Soliton and Fractals.2004,20(4),289-299.
[12]Constantin A and Escher J:Global existence and blow-up for a shallow water equation.Ann.Sc.Norm.Sup.Pisa.1998,26,303-328.
[13]Constantin A:Global existence of solutions and breaking waves for a shallow waterequation:a geometric approach.AnnJnst.Fourier (Grenoble).2000,50,321-362.
[14]Constantin A:The Hamitonian structure of the Camassa-Holm equation,Exposition.Math.1997,15,53-85.
[15]Constantin A,H.P.McKean:A shallow water equation on the circle,Comm.Pure Appl.Math.1999,52(8),949-982.
[16]Constantin A.,Strauss,W.A.:Stability of the Camassa-Holm solitons.J.Nonlinear Sci.2002,12,415-422.
[17]Constantin A:On the scattering problem for the Camassa-Holm equation.Proc.R.Soc.London A.2001,457,953-970.
[18]Lenells J:The Scattering approach for the Camassa-Holm equation.J.Nonliear Math.Phys. 2002,9(4),389-393.
[19]Beals R.,Sattinger D and Szmigielski J:Acoustic scattering and the extended KortewegdeVries hierarchy.Adv.Math.,1998,140,190-206.
[20]Constantin A and Escher J:Wave Breaking for Nonlinear Nonlocal Shallow Water Equations,Acta Mathematica.1998,181,229-243.
[21]Danchin R:A Few Remarks on the Camassa-Holm Equation,Differential and Integral Equations.2001,14,953-988.
[22]Guo BL,Liu ZR.:Peaked wave solutions of CH-r equation,Sci China (Ser..A).2003,33(4),325-337.
[23]Minying Tang,Chengxi Yang:Extension on peaked wave solutions of CH-r equation.Chaos,Solitons and Fractals.2004,20,815-825.
[24]J.Bona,R.Smith:The initial-value problem for the Korteweg-de Vries equation.Philos.Trans.Royal Soc.London Series A.1975,278,555-601.
[25]A.Degasperis,M.Proceci:Asymptotic integrability,in Symmetry and Perturbation Theory,edited by A.Degasperis and G.Gaeta,World Scientific(1999),23-37.
[26]P.G.Drazin and R.SJohnson,Solitions:an Introduction,Cambridge University press,Cambridge-New York,1989.
[27]H.P.Mckean:Integrable systems and algebraic curves.[J].GlobalAnalysis.Springer Lecture Notes in Mathematicas.1979,755,83-200.
[28]C.Kenig,G.Ponce,and L.Vega:Well-posedness and scattering results for the generalized Korteweg-de Veris equation via the contraction principle.[J].Comm.pure Appl.Math.1993,46,527-620.
[29]A.Constantin and J.Escher:Global existence and blow-up for a shallow water equation.[J].Annali Sc.Norm.Sup.Pisa.1998,26,303-328.
[30]Y.Li and P.Olver:Well-posedness and blow-up solutions for an integrable nonlinearly dispersive model wave equation.[J].J.Differential Equations.2000,162,27-63.
[31]A.Constantin:Global existence of solutions and breaking waves for a shallow water equations geometric approach.[J].Ann.Inst.Fourier(Grenoble).2000,50,321-362.
[32]A.Constantin and H.P.McKean:A shallow water equation on the circle.[J].Comm,Pure Appl.Math.1999,52,949-982.
[33]A.Degasperis,D.D.Holm,A.N.W.Hone:A New Integral Equation with Peakon Solutions[J].Theoretical and Mathematical Physics,2002,133,1463-1474.
[34]H.R.Dullin,G.A.Gottwald and D.D.Holm:Camassa-Holm,Korteweg-de Veris-5 and other asymptotically equivalent equations for shallow water waves.[J].Fluid Dynamics Research.2003,33,73-79.
[35]V.O.Vakhnenko,EJ.Parkes:Periodic and solitary-wave solutions of the Degasperis- Procesi equation.[JJ.Chaos Solitons Fractals.2004,20,1059-1073.
[36]Z.Yin:On the Cauchy problem for an integrable equation with peakon solutions.[J].Illinois J.Math.2003,47(1),649-666.
[37]Z.Yin:Global weak solutions to a new periodic integrable equation with peakon solutions.[J].J.Funct.Anal.2004,212,182-194.
[38]Z.Yin:Global solutions to a new integrable equation with peakons.[J].Indiana Univ.Math.J.2004,53,1189-1210.
[39]J.Lenells:Traveling wave solutions of the Degasperis-Procesi equation.[J].J.Math.Anal.Appl.2005,306,72-82.
[40]Y.Matsuno:Multisolition solutions of the Degasperis-Procesi equation and their peakon limit.[J].Invere Problems,2005,21,1553-1570.
[41]Constantine A.Popov:Perturbation theory for the doule Sine-Gordon equation.Wave Motion.2005,Vol 42,4,309-316.
[42]Mingling Wang,Xiangzheng Li:Exact solutions to the double Sine-Gordon equation.Chaos,Solitons and Fractals.2006,Vol 27,Issue 2,477-486.
[43]Shuimeng Yu,Weihua Yang,Jinxian Ruan,Lixin Tian:Solitary Wave Solutions to Approximate Fully Nonlinear Double Sine-Gordon Equation.International Journal of Nonlinear Science.2007,Vol 3,No3,163-169.
[44]Rosenau P,Hyman J.M.:Compactons.solitons with finite wavelength.[J],Phys.Rev.Lett.1993,70,564-567.
[45]Lixin Tian,Jiuli Yin:New peakon and multi-compacton solitary wave solutions of fully nonlinear Sine-Gordon equation.Chaos,Solitons and Fractals.2005,Vol 24,Issuel,353-363.
[46]Jiuli Yin,Lixin Tian:Peakon and Compacton Solutions for K (p q,\) Equation.International Journal of Nonlinear Science.2007,Vol 3,No.2,133-136.
[47]Lixin Tian,Xiuying Song:New peaked solitary wave solutions of the generalized Camassa-Holm equation.Chaos,Solitons and Fractals.2004,Vol 19,Issue 3,621 -637.
[48]Shuyi Zhang:Discontinous solutions to the Euler equations for a van der waals fluid.Appl.Math.Lett.2007,20,170-176.
[49]Lixin Tian,Shuimeng Yu,Meijie Ni:Noncontinuous Solitary Wave solutions for Double Sine-Gordon Equation.Chinese Physics Letters.2007,Vol.24,No.7,1791-1794.
[50]G.M.Coclite,H.Holden,K.H.Karlsen:Well-posedness for a parabolic-elliptic system,Discrete Contin.Dyn.Syst.2005,13,659-682.
[51]G.M.Coclite,H.Holden,K.H.Karlsen:Global weak solutions to a generalized hyperelasticrod wave equation,SIAM J.Math.Anal.2005,37,1044-1069.
[52]M.Fonte:Conservative solution of the Camassa-Holm equation on the real line,math.AP/0511549.
[53]H.Holden,X.Raynaud:Global conservative solutions of the Camassa-Holm equation-a Lagrangian point of view,Comm.Partial Differential Equations,in press.
[54]G.B.Folland:Real Analysis,seconded.,Wiley,New York,1999.
[55]H.Holden,X.Raynaud:Global conservative solutions of the generalized hyperelastic-rod wave equation,J.Differential Equations.2007,233,448-484.