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 α 1 ( t ) , β ( t ) and γ ( t ) are the dispersive, dissipative and line-damping coefficients, respectively, where t is the temporal variable. Bilinear forms, bilinear Backlund 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. Effects of α 1 ( t ) , β ( t ) and γ ( t ) on the solutions are discussed analytically and graphically. For the solitons, we find that α 1 ( 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, α 1 ( 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.