Abstract
By using solutions of an ordinary differential equation, an auxiliary equation method is described to seek exact solutions of variable-coefficient KdV-MKdV equation. As a result, more new exact nontravelling solutions, which include soliton solutions, combined soliton solutions, triangular periodic solutions, Jacobi elliptic function solutions, and combined Jacobi elliptic function solutions, for the KdV-MKdV equation are obtained. It is shown that the considered method provides a very effective, convenient, and powerful mathematical tool for solving many other nonlinear partial differential equations with variable coefficients in mathematical physics.
Highlights
Many important phenomena in various fields can be described by the nonlinear partial differential equations (NPDEs)
It is well known that NPDEs with variable coefficients are more realistic in various physical situations than their constant coefficients counterparts
Most of the above methods are related to the constant-coefficient NPDEs
Summary
Many important phenomena in various fields can be described by the nonlinear partial differential equations (NPDEs). Equation (1) can be used to describe the propagation of weakly nonlinear long waves in a KdV-typed medium by changing the coefficients of dispersion and nonlinear coefficients It includes the following three important equations:. − h (t) (2u + xux) , which models many important nonlinear phenomena, including shallow water waves, dust acoustic solitary structures. − h0 (t) (u + xux) , which models many important nonlinear phenomena, including shallow water waves, dust acoustic solitary structures in magnetized dusty plasmas, and ion acoustic waves in plasmas. + h (t) uxxx = 0, which is widely used in various branches of physics, such as plasma physics, fluid physics, and quantum field theory [24, 25] It describes a variety of wave phenomena in plasma and solid state.
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