The Cauchy problem and stability of solitary-wave solutions for RLW-KP-Type equations

Abstract

The Kadomtsev-Petviashvilli (KP) equation, (ut + ux + uux + uxxx )x + euyy = 0, (*) arises in various contexts where nonlinear dispersive waves propagate principally along the x-axis, but with weak dispersive effects being felt in the direction parallel to the y-axis perpendicular to the main direction of propagation. We propose and analyze here a class of evolution equations of the form (ut + ux, + upux + Lut)x + euyy = 0, (* *) which provides an alternative to Eq. (*) in the same way the regularized long-wave equation is related to the classical Korteweg-de Vries (KdV) equation. The operator L is a pseudo-differential operator in the x-variable, p ≥ 1 is an integer and e = ± 1. After discussing the underlying motivation for the class (* *), a local well-posedness theory for the initial-value problem is developed. With assumptions on L and p that include conditions appertaining to models of interesting physical phenomenon, the solutions defined locally in time t are shown to be smoothly extendable to the entire time-axis. In the particularly interesting case where L = -∂x2 and e = - 1, (*) possesses travelling-wave solutions u(x, y, t) = φc(x - ct, y) provided c > 1 and 0 < p < 4. It is shown here that these solitary waves are stable for 0 < 4/3 and c > I and for 4/3 <4 if c > (4p)/(4 + p). The paper concludes with commentary on extensions of the present theory to more than two space dimensions. © 2002 Elsevier Science (USA).