INVARIANT SUBSPACES AND GENERALIZED FUNCTIONAL SEPARABLE SOLUTIONS TO THE TWO-COMPONENT b-FAMILY SYSTEM

doi:10.1016/S0252-9602(16)30037-6

数学物理学报(英文版) ›› 2016, Vol. 36 ›› Issue (3): 753-764.doi: 10.1016/S0252-9602(16)30037-6

INVARIANT SUBSPACES AND GENERALIZED FUNCTIONAL SEPARABLE SOLUTIONS TO THE TWO-COMPONENT b-FAMILY SYSTEM

闫璐¹, 时振华², 王昊², 康静³

1. Xingzhi College, Xi'an University of Finance and Economics, Xi'an 710038, China;
2. School of Mathematics, Northwest University, Xi'an 710069, China;
3. Center for Nonlinear Studies and School of Mathematics, Northwest University, Xi'an 710069, China

收稿日期:2015-03-31 修回日期:2015-11-24 出版日期:2016-06-25 发布日期:2016-06-25
通讯作者: Jing KANG,E-mail:jingkang@nwu.edu.cn E-mail:jingkang@nwu.edu.cn
作者简介:Lu YAN,E-mail:xiaolu_4002@163.com;Zhenhua SHI,E-mail:andy_szh@163.com;Hao WANG,E-mail:610191181@qq.com
基金资助:
This work is supported by NSFC (11471260) and the Foundation of Shannxi Education Committee (12JK0850).

INVARIANT SUBSPACES AND GENERALIZED FUNCTIONAL SEPARABLE SOLUTIONS TO THE TWO-COMPONENT b-FAMILY SYSTEM

Lu YAN¹, Zhenhua SHI², Hao WANG², Jing KANG³

1. Xingzhi College, Xi'an University of Finance and Economics, Xi'an 710038, China;
2. School of Mathematics, Northwest University, Xi'an 710069, China;
3. Center for Nonlinear Studies and School of Mathematics, Northwest University, Xi'an 710069, China

Received:2015-03-31 Revised:2015-11-24 Online:2016-06-25 Published:2016-06-25
Contact: Jing KANG,E-mail:jingkang@nwu.edu.cn E-mail:jingkang@nwu.edu.cn
Supported by:
This work is supported by NSFC (11471260) and the Foundation of Shannxi Education Committee (12JK0850).

摘要/Abstract

摘要：

Invariant subspace method is exploited to obtain exact solutions of the two-component b-family system. It is shown that the two-component b-family system admits the generalized functional separable solutions. Furthermore, blow up and behavior of those exact solutions are also investigated.

关键词: invariant subspace, generalized conditional symmetry, generalized functional separable solution, Camassa-Holm equation, two-component b-family system

Abstract:

Key words: invariant subspace, generalized conditional symmetry, generalized functional separable solution, Camassa-Holm equation, two-component b-family system

中图分类号:

37K05

闫璐, 时振华, 王昊, 康静. INVARIANT SUBSPACES AND GENERALIZED FUNCTIONAL SEPARABLE SOLUTIONS TO THE TWO-COMPONENT b-FAMILY SYSTEM[J]. 数学物理学报(英文版), 2016, 36(3): 753-764.

Lu YAN, Zhenhua SHI, Hao WANG, Jing KANG. INVARIANT SUBSPACES AND GENERALIZED FUNCTIONAL SEPARABLE SOLUTIONS TO THE TWO-COMPONENT b-FAMILY SYSTEM[J]. Acta mathematica scientia,Series B, 2016, 36(3): 753-764.

参考文献

[1] Olver P J, Rosenau P. Tri-Hamiltonian duality between solitons and solitary-wave solutions having compact support. Phys Rev E, 1996, 53: 1900-1906
[2] Constantin A, Ivanov R I. On an integrable two-component Camassa-Holm shallow water system. Phys Lett A, 2008, 372: 7129-7132
[3] Camassa R, Holm D. An integrable shallow water equation with peaked solitons. Phys Rev Lett, 1993, 71: 1661-1664
[4] Fuchssteiner B, Fokas A. Symplectic structures, their Bäcklund transformations and hereditary symmetries. Physica D, 1981/1982, 4: 47-66
[5] Constantin A, Lannes D. The hydrodynamical relevance of the Camassa-Holm and Degasperis-Procesi equations. Arch Ration Mech Anal, 2009, 192: 165-186
[6] Chen M, Liu S Q, Zhang Y J. A two-component generalization of the Camassa-Holm equation and its solutions. Lett Math Phys, 2006, 75: 1-14
[7] Eschel J, Lechtenfeld O, Yin Z.Well-posedness and blow-up phenomena for the 2-component Camassa-Holm equation. Discrete Contin Dyn Syst Ser A, 2007, 19: 493-513
[8] Guan C X, Yin Z Y. Global existence and blow-up phenomena for an integrable two-component Camassa-Holm shallow water system. J Diff Equat, 2010, 248: 2003-2014
[9] Gui G L, Liu Y. On the global existence and wave-breaking criteria for the two-component Camassa-Holm system. J Funct Anal, 2010, 258: 4251-4278
[10] Yuen M. Self-similar blow up solutions to the two-component Camassa-Holm equations. J Math Phys, 2010, 51: 093524
[11] Fokas A S, Liu Q M. Asymptotic integrability of water waves. Phys Rev Lett, 1996, 77: 2347-2351
[12] Dullin R, Gottwald G, Holm D D. An integrable shallow water equation with linear and nonlinear dispersion. Phys Rev Lett, 2001, 87: 4501-4504
[13] Johnson R S. Camassa-Holm, Korteweg-de Vries and related models for water waves. J Fluid Mech, 2002, 455: 63-82
[14] Fu Y G, Liu Z R, Tang H. Non-uniform dependence on initial data for the modified Camassa-Holm equation on the line. Acta Math Sci, 2014, 34B(6): 1781-1794
[15] Constantin A. Existence of permanent and breaking waves for a shallow water equation: a geometric approach. Ann Inst Fourier (Grenoble), 2000, 50: 321-362
[16] Constantin A, Escher J. Wave breaking for nonlinear nonlocal shallow water equations. Acta Math, 1998, 181: 229-243
[17] Constantin A, Escher J. Well-posedness, global existence and blowup phenomena for a periodic quasi-linear hyperbolic equation. Comm Pure Appl Math, 1998, 51: 475-504
[18] Constantin A, Escher J. Global existence and blow-up for a shallow water equation. Ann Scuola Norm Sup Pisa, 1998, 26: 303-328
[19] Constantin A, Strauss W A. Stability of a class of solitary waves in compressible elastic rods. Phys Lett A, 2000, 270: 140-148
[20] Ito M. Symmetries and conservation laws of a coupled nonlinear wave equation. Phys Lett A, 1982, 91: 335-338
[21] Degasperis A, Procesi M. Asymptotic integrability//Degasperis A, Gaeta G. Symmetry and Perturbation Theory. World Scientific, 1999: 23-37
[22] Degasberis A, Holm D D, Hone A N W. A new integrable evolution equation with peakon solution. Theor Math Phys, 2002, 133: 1461-1472
[23] Liu Y, Yin Z Y. Global existence and blow-up phenomena for the Degasperis-Procesi equation. Comm Math Phys, 2006, 267: 801-820
[24] Yuen M. Self-similar blow up solutions to the Degasperis-Procesi shallow water system. Comm Non Sci Numer Simul, 2011, 16: 3463-3469
[25] Yuen M. Perturbed blow up solutions to the two-component Camassa-Holm equations. J Math Anal Appl, 2012, 390: 596-602
[26] Galaktionov V A, Svirshchevski S. Exact solutions and Invariant subspaces of Nonlinear Partial Differential Equations in Mechanics and Physics. London: Chapman and Hall, 2007
[27] Galaktionov V A. Invariant subspaces and new explicit solutions to evolution equations with quadratic nonlinearities. Proc R Soc Edinburgh, 1995, 125: 225-246
[28] Svirshchevski S. Invariant linear spaces and exact solutions of nonlinear evolution equations. J Non Math Phys, 1996, 3: 164-169
[29] Ji L N, Qu C Z. Conditional Lie-Bäcklund symmetries and invariant subspaces to nonlinear diffusion equations with convection and source. Stud Appl Math, 2013, 131: 266-301
[30] Zhu C R, Qu C Z. Maximal dimension of invariant subspaces admitted by nonlinear vector differential operators. J Math Phys, 2011, 52: 043507
[31] Zhdanov R Z. Conditional Lie-Bäcklund symmetries and reductions of evolution equations. J Phys A: Math Gen, 1995, 128: 3841-3850
[32] Fokas A S, Liu Q M. Nonlinear interaction of traveling waves of nonintegrable equations. Phys Rev Lett, 1994, 72: 3293-3296
[33] Qu C Z, Ji L N, Wang L Z. Conditional Lie-Bäcklund symmetries and sign-invarints to quasi-linear diffusion equations. Stud Appl Math, 2007, 119: 355-391
[34] Qu C Z. Group classification and generalized conditional symmetry reduction of the nonlinear diffusionconvection equation with a nonlinear source. Stud Appl Math, 1997, 99: 107-136
[35] Qu, C Z. Exact solutions to nonlinear diffusion equations obtained by a generalized conditional symmetry method. IMA J Appl Math, 1999, 62: 283-302
[36] Ji L N, Qu C Z, Ye Y J. Solutions and symmetry reductions of the n-dimensional nonlinear convectiondiffusion equations. IMA J Appl Math, 2010, 75: 17-55
[37] Basarab-Horwath P, Zhdanov R Z. Initial-value problems for evolutionary partial differential equations and higher-order conditional symmetries. J Math Phys, 2000, 42: 376-389
[38] Zhdanov R Z, Andreitsev A Y. Non-classical reductions of initial-value problems for a class of nonlinear evolution equations. J Phys A: Math Gen, 2000, 33: 5763-5781

INVARIANT SUBSPACES AND GENERALIZED FUNCTIONAL SEPARABLE SOLUTIONS TO THE TWO-COMPONENT b-FAMILY SYSTEM

INVARIANT SUBSPACES AND GENERALIZED FUNCTIONAL SEPARABLE SOLUTIONS TO THE TWO-COMPONENT b-FAMILY SYSTEM

PDF (PC)

摘要/Abstract

引用本文

使用本文

参考文献

相关文章 9

Metrics

本文评价

推荐阅读 4

[1]	肖杰胜, 曹广福. WANDERING SUBSPACES OF THE HARDY-SOBOLEV SPACES OVER Dⁿ[J]. 数学物理学报(英文版), 2016, 36(5): 1467-1473.
[2]	傅仰耿, 刘正荣, 唐昊. NON-UNIFORM DEPENDENCE ON INITIAL DATA FOR THE MODIFIED CAMASSA-HOLM EQUATION ON THE LINE[J]. 数学物理学报(英文版), 2014, 34(6): 1781-1794.
[3]	B. YOUSEFI, Sh. KHOSHDEL, Y. JAHANSHAHI. MULTIPLICATION OPERATORS ON INVARIANT SUBSPACES OF FUNCTION SPACES[J]. 数学物理学报(英文版), 2013, 33(5): 1463-1470.
[4]	B.Yousefi. UNIVERSAL INTERPOLATING SEQUENCES ON SPACES OF ANALYTIC FUNCTIONS[J]. 数学物理学报(英文版), 2011, 31(1): 68-72.
[5]	马璇,尹慧,金晶. GLOBAL ASYMPTOTICS TOWARD THE REREFACTION WAVES FOR A PARABOLIC-ELLIPTIC SYSTEM RELATED TO THE CAMASSA-HOLM SHALLOW WATER EQUATION[J]. 数学物理学报(英文版), 2009, 29(2): 371-390.
[6]	杨灵娥;纪艳珊;郭柏灵. THE RELATION OF TWO-DIMENSIONAL VISCOUS CAMASSA-HOLM EQUATIONS AND THE NAVIER-STOKES EQUATIONS[J]. 数学物理学报(英文版), 2009, 29(1): 65-73.
[7]	付一平; 郭柏灵. ALMOST PERIODIC SOLUTION OF ONE DIMENSIONAL VISCOUS CAMASSA-HOLM EQUATION[J]. 数学物理学报(英文版), 2007, 27(1): 117-124.
[8]	杜拴平, 侯晋川. CO-ISOMETRIC SOLUTIONS OF EQUATIONCU + U*C = 2D[J]. 数学物理学报(英文版), 2003, 23(2): 192-.
[9]	谭连生. THE DISTURBANCE LOCALIZATION PROBLEM FOR SINGULAR SYSTEMS[J]. 数学物理学报(英文版), 1995, 15(3): 241-246.