Stability of solitary waves for generalized <i>abcd</i>-Boussinesq system: the Hamiltonian case^*

Abstract The abcd -Boussinesq system is a model of two equations that can describe the propagation of small-amplitude long waves in both directions in the water of finite depth. Considering the Hamiltonian regimes, where the parameters b and d in the system satisfy <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mi>b</mml:mi> <mml:mo>=</mml:mo> <mml:mi>d</mml:mi> <mml:mo>&gt;</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> </mml:math> , small solutions in the energy space are globally defined. Then, a variational approach is applied to establish the existence and nonlinear stability of the set of solitary-wave solutions for the generalized abcb -Boussinesq system. The main point of the analysis is to show that the traveling-wave solutions of the generalized abcb -Boussinesq system converge to nontrivial solitary-wave solutions of the generalized Korteweg–de Vries equation. Moreover, if p is the exponent of the nonlinear terms for the generalized abcb -Boussinesq system, then the nonlinear stability of the set of solitary-waves is obtained for any p with <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mn>1</mml:mn> <mml:mtext>⩽</mml:mtext> <mml:mi>p</mml:mi> <mml:mo>&lt;</mml:mo> <mml:msub> <mml:mi>p</mml:mi> <mml:mn>0</mml:mn> </mml:msub> </mml:mrow> </mml:math> where p 0 is strictly larger than 4, while it has been known that the critical exponent for the stability of solitary waves of the generalized KdV equations is equal to 4.