Perturbation of a Simple Wave in a System with Dissipation

A differential equation that models the motion of a domain wall is considered. For this equation, in the case of constant coefficients, there exists a solution in the form of a travelling wave which describes the transition from one equilibrium to another. For an equation with slowly varying coefficients, we construct an asymptotic one-phase solution. The phase is found from the Hamilton–Jacobi equation whose coefficients are taken from the asymptotics of the unperturbed wave at infinity. The asymptotic construction is based on the requirement that the first correction is small as compared to the leading term uniformly over a wide range of independent variables.