Existence and orbital stability of standing waves for nonlinear Schrödinger systems

2015/12/30 by Tianxiang Gou, Louis Jeanjean, Gou, Tianxiang +1 · 5 citations
Mathematics · Physics and Astronomy · #Advanced Mathematical Physics Problems #Nonlinear Photonic Systems #Spectral Theory in Mathematical Physics #math.AP

paper · pdf · doi:10.48550/arxiv.1512.08952

arxiv created 2016/02/27 · arxiv updated 2016/03/01

Abstract

In this paper we investigate the existence of solutions in H¹(R^N) × H¹(R^N) for nonlinear Schrödinger systems of the form \ \beginaligned -Δu₁ = λ₁ u₁ + μ₁ |u₁|^{p₁ -2}u₁ + r₁β|u₁|^r₁-2u₁|u₂|^r₂,
-Δu₂ = λ₂ u₂ + μ₂ |u₂|^{p₂ -2}u₂ + r₂ β|u₁|^r₁|u₂|^{r₂ -2}u₂, \endaligned . under the constraints ∫_R^N|u₁|² dx = a₁>0, ∫_R^N|u₂|² dx = a₂>0. Here N ≥ 1, β>0, μ_i >0, r_i >1, 2 <p_i < 2 + (4)/(N) for i=1,2 and r₁ + r₂ < 2 + (4)/(N). This problem is motivated by the search of standing waves for an evolution problem appearing in several physical models. Our solutions are obtained as constrained global minimizers of an associated functional. Note that in the system λ₁ and λ₂ are unknown and will correspond to the Lagrange multipliers. Our main result is the precompactness of the minimizing sequences, up to translation, and as a consequence we obtain the orbital stability of the standing waves associated to the set of minimizers.

Citations

• Orbital stability of standing waves for some nonlinear Schr�dinger equations

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