Bilinear forms, bilinear Bäcklund transformation, soliton and breather interactions of a damped variable-coefficient fifth-order modified Korteweg–de Vries equation for the surface waves in a strait or large channel

Abstract

In this paper, we investigate a damped variable-coefficient fifth-order modified Korteweg–de Vries equation for the small-amplitude surface waves in a strait or large channel of slowly-varying depth and width and non-vanishing vorticity, in which α1(t), β(t) and γ(t) are the dispersive, dissipative and line-damping coefficients, respectively, where t is the temporal variable. Bilinear forms, bilinear Bäcklund transformation and multi-soliton solutions are constructed via the Hirota bilinear method under some variable-coefficient constraints. Based on those multi-soliton solutions, multi-pole, breather and hybrid solutions are derived. Effect of α1(t), β(t) and γ(t) on the solutions is discussed analytically and graphically. For the solitons, we find that α1(t) and β(t) are related to the velocities and characteristic lines, and the amplitudes depend on γ(t). For the multi-pole and breather solutions, α1(t) and β(t) influence the center trajectories of the solutions, while γ(t) influences the amplitudes. Hybrid solutions composed of the breathers and solitons are worked out and discussed graphically.