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

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-Huxrey, and Korteweg-de Vries-Burgers-Huxl ey equations, as well as equations in the generic form u(t) + P(u) u(x) + v u(xx) - delta u(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 relev ant equations in physics. We focus on the analysis of the role of diffusion , dispersion, nonlinear effects, and parity of the polynomials to the prope rties of the solutions, particularly their velocity of propagation. In addi tion, we show that, for some appropriate choices, these equations can be ma pped onto equations of motion of relativistic (1 + 1)-dimensional phi(4) an d phi(6) field theories of real scalar fields. Systems of two coupled nonli near equations are also considered. (C) 1999 Elsevier Science B.V.