Mathematics

During this seminar, I will present the results of a recently submitted paper, wherein we consider a family of generalized Korteweg-de Vries equations and study the linear modulational instability of small amplitude traveling waves solutions. Under explicit non-degeneracy conditions on the dispersion relation, we completely describe the spectrum near the origin of the linearized operator at such solutions and prove that the unstable spectrum (when present) is composed by branches depicting always a closed figure 8. We apply our abstract theorem to several equations such as the Whitham, the gravity-capillary Whitham and the Kawahara equations, confirming that the unstable spectrum of the corresponding linearized operators exhibits a figure 8 instability, as it was observed before only numerically. Our method of proof uses a symplectic version of Kato’s theory of similarity transformation to reduce the problem to determine the eigenvalues of a 3x3 complex Hamiltonian and reversible matrix. Then, via a block-diagonalization procedure, we conjugate such matrix into a block-diagonal one composed by a 2x2 Hamiltonian and reversible matrix, describing the unstable spectrum, and a single purely imaginary element describing the stable eigenvalue.