A note on the uniqueness properties of solutions for the Schrödinger-Korteweg de Vries system

2025/07/02 by Eddye Bustamante, Bustamante, Eddye, José Jiménez Urrea +3
Mathematics · #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #Navier-Stokes equation solutions #Nonlinear Partial Differential Equations

paper · pdf · doi:10.48550/arxiv.2507.01733

openalex publication_date 2025/07/02 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this work we prove that if (u_i,v_i), i=1,2, are smooth enough solutions of the coupled Schrödinger-Korteweg-de Vries system . i u_t+∂_x² u · amp;\hspace-2mm=βuv - |u|² u,
∂_t v + ∂_x³ v · amp;\hspace-2mm=γ∂_x |u|²-\frac12∂_x (v²) \ with appropriate decay at infinity such that at two different times t₀=0 and t₁=1 satisfy that u₁(0)-u₂(0),u₁(1)-u₂(1),v₁(0)-v₂(0),v₁(1)-v₂(1)∈ H¹(e^ax²dx), for a>0 big enough, then u₁=u₂ and v₁=v₂. (Let us recall that f∈ H¹(e^ax² dx) iff f∈ L²(e^ax²dx) and ∂_x f∈ L²(e^ax²dx)).

Citations

• Sharp local well-posedness for the Schrödinger-Korteweg-de Vries system
• Dispersive blow-up and persistence properties for the\n Schr "odinger-Korteweg-de Vries system
• On the support of solutions to the Zakharov-Kuznetsov equation
• On uniqueness properties of solutions of the k-generalized KdV equations

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• Normalized solutions for nonlinear Schrödinger systems with special mass-mixed terms: The linear couple case