Stability and instability of periodic travelling wave solutions for the critical Korteweg–de Vries and nonlinear Schrödinger equations

In this paper we establish new results about the existence, stability, and instability of periodic travelling wave solutions related to the critical Korteweg–de Vries equation u t + 5 u 4 u x + u x x x = 0 , and the critical nonlinear Schrödinger equation i v t + v x x + | v | 4 v = 0 . The periodic travelling wave solutions obtained in our study tend to the classical solitary wave solutions in the infinite wavelength scenario. The stability approach is based on the theory developed by Angulo & Natali in [J. Angulo, F. Natali, Positivity properties of the Fourier transform and the stability of periodic travelling wave solutions, SIAM J. Math. Anal. 40 (2008) 1123–1151] for positive periodic travelling wave solutions associated to dispersive evolution equations of Korteweg–de Vries type. The instability approach is based on an extension to the periodic setting of arguments found in Bona & Souganidis & Strauss [J.L. Bona, P.E. Souganidis, W.A. Strauss, Stability and instability of solitary waves of Korteweg–de Vries type, Proc. Roy. Soc. London Ser. A 411 (1987) 395–412]. Regarding the critical Schrödinger equation stability/instability theories similar to the critical Korteweg–de Vries equation are obtained by using the classical Grillakis & Shatah & Strauss theory in [M. Grillakis, J. Shatah, W. Strauss, Stability theory of solitary waves in the presence of symmetry II, J. Funct. Anal. 94 (1990) 308–348; M. Grillakis, J. Shatah, W. Strauss, Stability theory of solitary waves in the presence of symmetry I, J. Funct. Anal. 74 (1987) 160–197]. The arguments presented in this investigation have prospects for the study of the stability of periodic travelling wave solutions of other nonlinear evolution equations.