Existence of traveling waves for a fourth order Schrödinger equation with mixed dispersion in the Helmholtz regime
Year: 2026
Type: article
Abstract: In this paper, we study the existence of traveling waves for a fourth order Schrödinger equations with mixed dispersion, that is, solutions to $$\Delta ^2 u +\beta \Delta u - Q(-i V \nabla ) u +\alpha u =|u|^{p-2} u,\ \text {in } \ \mathbb {R}^N,\ N\ge 2,$$ Δ 2 u + β Δ u - Q ( - i V ∇ ) u + α u = | u | p - 2 u , in R N , N ≥ 2 , where the Q term represents a lower order symmetry breaking operator. We consider this equation in the Helmholtz regime, that is, when the Fourier symbol P of the operator is sign-changing and we assume that the zero set of the symbol is a manifold M . Under suitable assumptions, we prove the existence of solution using the dual method of Evequoz and Weth provided that $$p\in (p_1, 2N/(N-4)_+)$$ p ∈ ( p 1 , 2 N / ( N - 4 ) + ) , where the real number $$p_1$$ p 1 depends on the number of principal curvature of M staying bounded away from 0. We also obtain estimates on the Green’s function of our operator and a $$L^p - L^q$$ L p - L q resolvent-type estimate which can be of independent interest and can be extended to other operators.
Source: Mathematische Annalen
Authors: Jean-Baptiste Casteras, Juraj Földes
Institutions: University of Lisbon, University of Virginia
Language: English
Subfield: Applied Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: hybrid
Estimated APC paid: $3,190
OpenAlex ID: W7128788802