The Generalized Fractional Benjamin-Bona-Mahony Equation: Analytical and Numerical Results

TL;DR

This paper investigates the generalized fractional Benjamin-Bona-Mahony equation, establishing local existence, uniqueness, and solitary wave solutions, and explores their numerical evolution and the interplay between nonlinearity and fractional dispersion.

Contribution

It provides the first analytical proof of local solutions and conditions for solitary waves for the gfBBM equation, along with numerical methods for their simulation.

Findings

Existence and uniqueness of solutions proved

Conditions for solitary wave solutions derived

Numerical experiments reveal the relation between nonlinearity and fractional dispersion

Abstract

The generalized fractional Benjamin-Bona-Mahony (gfBBM) equation models the propagation of small amplitude long unidirectional waves in a nonlocally and nonlinearly elastic medium. The equation involves two fractional terms unlike the well-known fBBM equation. In this paper, we prove local existence and uniqueness of the solutions for the Cauchy problem. The sufficient conditions for the existence of solitary wave solutions are obtained. The Petviashvili method is proposed for the generation of the solitary wave solutions and their evolution in time is investigated numerically by Fourier spectral method. The efficiency of the numerical methods is tested and the relation between nonlinearity and fractional dispersion is observed by various numerical experiments.