2023/06/29 by Filho, Roberto de A. Capistrano, Quintero, Jose Raul, Sun, Shu-Ming
#35B35 #35Q35 #76B25 #Analysis of PDEs (math.AP) #FOS: Mathematics
paper · doi:10.48550/arxiv.2306.17335
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 b=d>0, 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 0 < p < p0 where p0 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.