Exact kink solitons in the presence of diffusion, dispersion, and polynomial nonlinearity

Abstract We describe exact travelling-wave kink soliton solutions in some classes of nonlinear partial differential equations, such as generalized Korteweg-de Vries-Burgers, Korteweg-de Vries-Huxley, and Korteweg-de Vries-Burgers-Huxley equations, as well as equations in the generic form u t + P ( u ) u x + vu xx − δu xxx = A ( u ), with polynomial functions P ( u ) and A ( u ) of u = u ( x , t ), whose generality allows the identification with a number of relevant equations in physics. We focus on the analysis of the role of diffusion, dispersion, nonlinear effects, and parity of the polynomials to the properties of the solutions, particularly their velocity of propagation. In addition, we show that, for some appropriate choices, these equations can be mapped onto equations of motion of relativistic (1 + 1)-dimensional φ 4 and φ 6 field theories of real scalar fields. Systems of two coupled nonlinear equations are also considered.