Nonlinear Helmholtz equations with sign-changing diffusion coefficient

Abstract

In this talk, we are interested in the combine effect of metamaterial and Kerr non-linearity. More precisely, we study nonlinear Helmholtz equations with sign-changing diffusion coefficients on bounded domains of the form -div (σ(x) ∇ u) - λ u = u^3. Using weak 𝚃-coercivity theory, we can establish the existence of an orthonormal basis of eigenfunctions of the linear part -div (σ(x) ∇ u). Then, all eigenvalues are proved to be bifurcation points, and we investigate the bifurcating branches both theoretically and numerically. As a fundamental example, we look at some one-dimensional model, we obtain the existence of infinitely many bifurcating branches that are mutually disjoint, unbounded, and consist of solutions with a fixed nodal pattern.