Conservation Laws for a Model with both Cubic and Quadratic Nonlinearity

Abstract

In this paper, the conservation laws for a model with both quadratic and cubic nonlinearity \begin{eqnarray*}m_{t}=bu_{x}+\frac{1}{2}a\left[ \left( u^{2}-u_{x}^{2}\right) m\right] _{x}+%\frac{1}{2}c\left( 2m\cdot u_{x}+m_{x}\cdot u\right) ;\text{ \ \ }m=u-u_{xx}\end{eqnarray*}%are considered for the six cases of coefficients. By using a variational derivative approach, conservation laws were constructed. The computations to derive multipliers and conservation law fluxes are conducted by using a Maple-based package which is called GeM.