The Traveling Wave Solutions and Their Bifurcations for the BBM-Like B(m, n) Equations

We investigate the traveling wave solutions and their bifurcations for the BBM-like equations by using bifurcation method and numerical simulation approach of dynamical systems. Firstly, for BBM-like equation, we obtain some precise expressions of traveling wave solutions, which include periodic blow-up and periodic wave solution, peakon and periodic peakon wave solution, and solitary wave and blow-up solution. Furthermore, we reveal the relationships among these solutions theoretically. Secondly, for BBM-like equation, we construct two periodic wave solutions and two blow-up solutions. In order to confirm the correctness of these solutions, we also check them by software Mathematica.

[1]  Jibin Li,et al.  New explicit and exact solutions for a system of variant RLW equations , 2008, Appl. Math. Comput..

[2]  Liu Zhengrong,et al.  Trigonometric Function Periodic Wave Solutions and Their Limit Forms for the KdV-Like and the PC-Like Equations , 2011 .

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

[4]  M. A. Abdou.  The extended F-expansion method and its application for a class of nonlinear evolution equations , 2007 .

[5]  Abdul-Majid Wazwaz,et al.  Analytic study on nonlinear variants of the RLW and the PHI-four equations , 2007 .

[6]  Shang Ya-dong.  Explicit and exact special solutions for BBM-like B(m, n) equations with fully nonlinear dispersion , 2005 .

[7]  T. Fan,et al.  New applications of developed Jacobi elliptic function expansion methods , 2005 .

[8]  D. Peregrine.  Calculations of the development of an undular bore , 1966, Journal of Fluid Mechanics.

[9]  Jibin Li,et al.  TRAVELING WAVE SOLUTIONS FOR A CLASS OF NONLINEAR DISPERSIVE EQUATIONS , 2002 .

[10]  Ming Song,et al.  Application of the (G'/G)-expansion method to (3 +1)-dimensional nonlinear evolution equations , 2010, Comput. Math. Appl..

[11]  Ş. Kuru.  Traveling wave solutions of the BBM-like equations , 2009 .

[12]  Xiangzheng Li,et al.  Extended F-expansion method and periodic wave solutions for the generalized Zakharov equations , 2005 .

[13]  Shaoyong Li,et al.  Some Explicit Expressions and Interesting Bifurcation Phenomena for Nonlinear Waves in Generalized Zakharov Equations , 2013 .

[14]  Rui Liu.  Some new results on explicit traveling wave solutions of $K(m, n)$ equation , 2010 .

[15]  HighWire Press.  Philosophical Transactions of the Royal Society of London , 1781, The London Medical Journal.

[16]  J. Bona,et al.  Model equations for long waves in nonlinear dispersive systems , 1972, Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences.

[17]  Ming Song,et al.  New Exact Solutions for the $(2+1)$-Dimensional Broer-Kaup-Kupershmidt Equations , 2010 .

[18]  D. Peregrine.  Long waves on a beach , 1967, Journal of Fluid Mechanics.

[19]  Yong Liang,et al.  The Explicit nonlinear wave solutions and their bifurcations of the Generalized Camassa-Holm equation , 2011, Int. J. Bifurc. Chaos.

[20]  Mingliang Wang,et al.  The (G' G)-expansion method and travelling wave solutions of nonlinear evolution equations in mathematical physics , 2008 .

[21]  Bo Jiang,et al.  Bifurcation of traveling wave solutions for the BBM-like B(2, 2) equation , 2012, Appl. Math. Comput..

[22]  Ş. Kuru.  Compactons and kink-like solutions of BBM-like equations by means of factorization , 2008, 0810.4166.

[23]  Jibin Li,et al.  Smooth and non-smooth traveling waves in a nonlinearly dispersive equation , 2000 .

[24]  Zuntao Fu,et al.  JACOBI ELLIPTIC FUNCTION EXPANSION METHOD AND PERIODIC WAVE SOLUTIONS OF NONLINEAR WAVE EQUATIONS , 2001 .