The bifurcation and peakon for K(2, 2) equation with osmosis dispersion

Abstract In this paper, the qualitative analysis methods of a dynamical system are used to investigate the peaked wave solutions of K(2, 2) equation with osmosis dispersion. The phase portrait bifurcation of the traveling wave system corresponding to the equation is given. The explicit expressions of the peaked solitary wave solution and the periodic cusp wave solution are obtained by using the portraits. The graph of the solution is given with the numerical simulation.

[1]  Hideaki Ujino,et al.  Integrable semi-discretization of the coupled nonlinear Schr\ , 1999, solv-int/9903013.

[2]  Philip Rosenau,et al.  On nonanalytic solitary waves formed by a nonlinear dispersion , 1997 .

[3]  L. Tian,et al.  The bifurcation and peakon for Degasperis–Procesi equation , 2006 .

[4]  Tifei Qian,et al.  Peakons and periodic cusp waves in a generalized Camassa–Holm equation , 2001 .

[5]  Philip Rosenau,et al.  On a class of nonlinear dispersive-dissipative interactions , 1997 .

[6]  B. Guo,et al.  Periodic cusp wave solutions and single-solitons for the b-equation , 2005 .

[7]  Lixin Tian,et al.  On the Well-Posedness Problem and the Scattering Problem for the Dullin-Gottwald-Holm Equation , 2005 .

[8]  Abdul-Majid Wazwaz,et al.  A reliable treatment of the physical structure for the nonlinear equation K(m, n) , 2005, Appl. Math. Comput..

[9]  P. T. Dinda,et al.  Breather compactons in nonlinear Klein-Gordon systems. , 1999, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[10]  Lixin Tian,et al.  Backlund Transformation and N-soliton-like Solutions to the Combined KdV-Burgers Equation with Variable Coefficients , 2006 .

[11]  B. Dey,et al.  Compacton solutions for a class of two parameter generalized odd-order Korteweg-de Vries equations , 1998 .

[12]  M. S. Ismail,et al.  A numerical study of compactons , 1998 .

[13]  Abdul-Majid Wazwaz,et al.  Exact special solutions with solitary patterns for the nonlinear dispersive K(m,n) equations , 2002 .

[14]  Boling Guo,et al.  Peaked wave solutions of CH-r equation , 2003 .

[15]  Tian Li-xin,et al.  The concave or convex peaked and smooth soliton solutions of Camassa-Holm equation , 2002 .

[16]  Abdul-Majid Wazwaz,et al.  General solutions with solitary patterns for the defocusing branch of the nonlinear dispersive K(n, n) equations in higher dimensional spaces , 2002, Appl. Math. Comput..

[17]  G. Afrouzi,et al.  On Adomian Decomposition Method for Solving Reaction Diffusion Equation , 2006 .

[18]  L. Tian,et al.  On the Well-posedness Problem for the Generalized Degasperis-Procesi Equation , 2006 .

[19]  Cooper,et al.  One-parameter family of soliton solutions with compact support in a class of generalized Korteweg-de Vries equations. , 1993, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[20]  Shepard,et al.  Solitary waves in a class of generalized Korteweg-de Vries equations. , 1993, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[21]  M. A. López-Marcos,et al.  A Finite Difference Scheme for the K(2, 2) Compacton Equation , 1995 .

[22]  M. S. Ismail.  A finite difference method for korteweg-de vries like equation with nonlinear dispersion , 2000, Int. J. Comput. Math..

[23]  Abdul-Majid Wazwaz,et al.  General compactons solutions for the focusing branch of the nonlinear dispersive K(n, n) equations in higher-dimensional spaces , 2002, Appl. Math. Comput..

[24]  Abdul-Majid Wazwaz,et al.  New solitary-wave special solutions with compact support for the nonlinear dispersive K(m, n) equations , 2002 .

[25]  L. Tian,et al.  Stability of multi-compacton solutions and Backlund transformation in K(m,n,1) , 2004 .

[26]  S. Lou,et al.  Painlevé integrability of two sets of nonlinear evolution equations with nonlinear dispersions , 1999 .

[27]  Abdul-Majid Wazwaz,et al.  Explicit travelling wave solutions of variants of the K(n, n) and the ZK(n, n) equations with compact and noncompact structures , 2006, Appl. Math. Comput..

[28]  Hyman,et al.  Compactons: Solitons with finite wavelength. , 1993, Physical review letters.

[29]  P. Rosenau,et al.  Nonlinear dispersion and compact structures. , 1994, Physical review letters.

[30]  Abdul-Majid Wazwaz,et al.  Compactons dispersive structures for variants of the K(n,n) and the KP equations , 2002 .

[31]  Zheng-rong Liu,et al.  Peaked wave solutions of Camassa–Holm equation , 2004 .