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Existence of solitary waves and periodic waves for a perturbed generalized BBM equation. (English) Zbl 1358.34051

This paper considers a perturbed generalized Benjamin-Bona-Mahony (BBM) equation of the form: \[ (u^2)_t+(u^3)_x+u_{xxx}+\varepsilon(u_{xx}+u_{xxxx})=0, \] where \(\varepsilon>0\) is a perturbation parameter. The existence of solitary waves and periodic waves for this kind of perturbed generalized BBM equation is established by using geometric singular perturbation theory. Attention goes to perturbations of the Hamiltonian vector field with an elliptic Hamiltonian of degree four, exhibiting a cuspidal loop. It is proven that the wave speed \(c_0(h)\) is decreasing for \(h\in [0, 1/12]\) by analyzing the ratio of abelian integrals (\(h\) being a parameter describing the level curves of the Hamiltonian). The upper and lower bounds of the limit wave speed are given. Moreover, the relation between the wave speed and the wavelength of traveling waves is obtained.

MSC:

34C60 Qualitative investigation and simulation of ordinary differential equation models
34C25 Periodic solutions to ordinary differential equations
34C27 Almost and pseudo-almost periodic solutions to ordinary differential equations
35C07 Traveling wave solutions
35C08 Soliton solutions
34E20 Singular perturbations, turning point theory, WKB methods for ordinary differential equations
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