Analytical models of stationary nonlinear gravitational waves

Abstract

Euler’s equations with standard boundary conditions for the problem of potential surface waves of an arbitrary amplitude in a homogeneous liquid layer with a flat bottom are converted into the new system, including integral and differential equations for the of the potential and its time derivative near the surface. The basic formula of the theory of infinitesimal waves, paired Korteweg-de Vries (KdV) and Kadomtsev− Petviashvili (KP) equations, the envelope Zakharov−Shabat soliton follows from the system in limiting case. The resulting generalized equation, unlike traditional KdFand KP-equations is suitable for the description of waves on the surface of the initially quiescent fluid. A new exact solutions for gravity waves in a deep water, expressed in terms of complex Lambert’s functions are constructed.