Traveling waves for a quasilinear wave equation

Abstract

We consider a 2+1 dimensional wave equation appearing in the context of polarized waves for the nonlinear Maxwell equations. The equation is quasilinear in the time derivatives and involves two material functions $V$ and $\Gamma$. We prove the existence of traveling waves which are periodic in the direction of propagation and localized in the direction orthogonal to the propagation direction. Depending on the nature of the nonlinearity coeffcient $\Gamma$ we distinguish between two cases: (a) $\Gamma\in L^\infty$ being regular and (b) $\Gamma=\gamma\delta_0$ being a multiple of the delta potential at zero. For both cases we use bifuraction theory to prove the existence of nontrivial small-amplitude solutions. One can regard our results as a persistence result which shows that guided modes known for linear wave-guide geometries survive in the presence of a nonlinear constitutive law. Our main theorems are derived under a set of conditions on the linear wave operator. They are subsidised by explicit examples for the coefficients $V$ in front of the (linear) second time derivative for which our results hold.