ON A CUBIC NONLINEAR EQUATION MODEL ARISING IN SHALLOW WATER THEORY

Abstract

The shallow water waves theory produces numerous integrable equations with cubic non- linearity as asymptotic models. We began our work by formally deriving a model equation for the free surface elevation η with higher-order terms from shallow water in the Euler equation for an incompressible fluid with the simplest bottom and surface conditions. This model equation is truncated at the order O(ε3,εμ) and contains higher-order terms, which are useful for deriving a class of unidirectional wave equations including cubic nonlinear terms. Next, we derived an equation with cubic nonlinearity as the asymptotic method from the classical shallow-water theory by employing suitable scalings, appropriate asymptotic expansions truncating, and a particular Kodama transformation to expand η in terms of u and its derivatives. This equation is relates to several different crucial shallow waterequations, including the CH, mCH, and Novikov types. Last, we analyzed a special case of our approximate equation called mCH-Novikov equation by applying the method of characteristics by using conserved quantities to arrive at a Riccati-type differential inequality. This proved that the wave-breaking phenomenon of this equation is the curvature blow-up.