(Non)linear instability of periodic traveling waves: Klein–Gordon and KdV type equations

Abstract

We prove the existence and nonlinear instability of periodic traveling wave solutions for the critical one-dimensional Klein–Gordon equation. We also establish a linear instability criterium for a KdV type system. An application of this approach is made to obtain the linear/nonlinear instability of vector cnoidal wave profiles. Finally, via a theoretical and numerical approach we show the linear stability or instability of periodic positive and sign changing waves, respectively, for the critical Korteweg–de Vries equation.

Keywords

• linear stability
• (non)linear instability
• critical klein–gordon equation
• critical kdv
• kdv type models
• 35q51
• 35b35
• 35b10
• 35c07