Shock waves in one-dimensional models with cubic nonlinearity

Shock waves are described qualitatively for a class of one-dimensional models with cubic nonlinearity (of the type of the modified Korteweg–de Vries equation): $u_t-6u^2u_x+\gamma u_{xxx}=\nu u_{xx}$. Both the integrable and the nonintegrable case are considered. The behavior of a shock wave in the limit $t \to \infty$ is considered.