The Cauchy Problem and Stability of Solitary-Wave Solutions for RLW–KP-Type Equations,

The Kadomtsev–Petviashvilli (KP) equation,(ut+ux+uux+uxxx)x+ uyy=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+ uyy=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 =±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 =−1, (*) possesses travelling-wave solutions u(x,y, t)=πc(x−ct,y) provided c>1 and 01 and for if c>(4p)/(4+p). The paper concludes with commentary on extensions of the present theory to more than two space dimensions.