The Replacement Rule for Nonlinear Shallow Water Waves

Abstract

When a $(1+1)$-dimensional nonlinear PDE in real function $\eta (x,t)$ admits localized traveling solutions we can consider $L$ to be the average width of the envelope, $A$ the average value of the amplitude of the envelope, and $V$ the group velocity of such a solution. The replacement rule (RR or nonlinear dispersion relation) procedure is able to provide a simple qualitative relation between these three parameters, without actually solve the equation. Examples are provided from KdV, C-H and BBM equations, but the procedure appears to be almost universally valid for such $(1+1)$-dimensional nonlinear PDE and their localized traveling solutions \cite{3}.

12 pages, 1 figure

References in corpus (1)

• An Integrable Shallow Water Equation with Linear and Nonlinear Dispersion