Abstract
Under investigation in this paper is a (3 + 1)‐dimensional variable‐coefficient generalized shallow water wave equation. The exact lump solutions of this equation are presented by virtue of its bilinear form and symbolic computation. Compared with the solutions of the previous cases, these solutions contain two inhomogeneous coefficients, which can show some interesting nonautonomous characteristics. Three types of dispersion coefficients are considered, including the periodic, exponential, and linear modulations. The corresponding nonautonomous lump waves have different characteristics of trajectories and velocities. The periodic fission and fusion interaction between a lump wave and a kink soliton is discussed graphically.
Highlights
Under investigation in this paper is a (3 + 1)-dimensional variable-coe cient generalized shallow water wave equation. e exact lump solutions of this equation are presented by virtue of its bilinear form and symbolic computation
We find that the variable coefficients α1(t) and α2(t) do not affect the width and amplitude of the lump wave
We have studied the (3 + 1)-dimensional variable-coefficient generalized shallow water wave equation, which characterizes the flow below a pressure surface in a fluid. rough the Hirota method, we have obtained nonautonomous lump solutions for equation (2)
Summary
Through symbolic computation, we investigate the nonautonomous characteristics of the lump solution of equation (2). Because of the periodic modulation, the range of motion of the lump wave along the x-axis (y-axis) is confined to (− 20, 20) [(− 10, 10)]. The velocity of the lump wave periodically varies with time. It is obvious that the trajectory of the lump wave is a half-line, which is different from the case of periodic modulation. As time increases (t < 0), the velocity of the wave decreases gradually. Compared with the previous cases, we discover that the velocity of the lump wave varies with time linearly
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