Travelling wave solutions to a perturbed Korteweg-de Vries equation
Traveling wave solutions are considered for the equation which may be regarded as a perturbed Korteweg-de Vries (KdV) equation, or a generalized Kuramoto-Sivashinsky equation: \[ u_ t+ uu_ x+ u_{xxx}+ \varepsilon (u_{xx}+ u_{xxxx})=0, \] where \(\varepsilon\) is a positive parameter. This equation describes flow of a thin layer of viscous liquid along an inclined plane, and finds a number of other applications. The traveling wave solution is looked for in the form \(u=u(x-ct)\), where \(c\) is a velocity. Substitution of this expression into the underlying equation leads to a fourth-order ordinary differential equation for the function \(u(z)\), where \(z\equiv x-ct\). The velocity \(c\) enters the latter equation as a parameter. The solutions sought for satisfy the periodic boundary condition \(u(z)= u(z+L)\) with an arbitrary period \(L\). Previously, it was demonstrated by means of a formal asymptotic procedure, treating \(\varepsilon\) as a small parameter, that for each period \(L\) there exists a single traveling wave solution with a certain velocity \(c(L)\). The procedure was based on using the well-known exact periodic cnoidal-wave solutions of the KdV equation as the zeroth-order approximation. The important circumstance is that there is a one- parametric family of the cnoidal-wave solutions for the KdV equation subject to the periodic boundary condition, the velocity being an arbitrary parameter. The first approximation of the formal perturbation theory selects a single value of the velocity for a given period \(L\). In the present work, using essentially the same idea, the existence of a periodic solution with a single value of the velocity is proved rigorously. It is also proved that in the limit \(\varepsilon\to 0\), the solution uniformly converges to the cnoidal wave with the same value of the velocity.
one-parametric family of cnoidal-wave solutions
generalized Kuramoto- Sivashinsky equation
fourth-order ordinary differential equation
scientific article; zbMATH DE number 681280