Stability of Periodic Travelling Wave Solutions for the Korteweg - de Vries Equation

This paper is concerned with nonlinear stability properties of periodic travelling waves solutions of the classical Korteweg - de Vries equation,\[\label{kdv} u_t+uu_x+u_{xxx}=0, \ \ x,t\in \mathbb R.\]It is shown the existence of a nontrivial smooth curve of periodic travelling wave solutions depending on the classical Jacobian elliptic functions. We find positive cnoidal wave solutions. Then we prove, by using the framework established in \cite{GSS1} by Grillakis, Shatah and Strauss, the nonlinear stability of the cnoidal wave solutions in the space $H^1_{per}([0,L])$.