[Paper Review] Bifurcation from semi-trivial standing waves and ground states for a system of nonlinear Schrödinger equations

Mathieu Colin, Masahito Ohta|arXiv (Cornell University)|Feb 8, 2011

Advanced Mathematical Physics Problems17 references4 citations

TL;DR

This paper investigates bifurcation from semi-trivial standing waves and ground states in a two-component nonlinear Schrödinger system modeling Raman amplification in plasmas. Using variational methods and bifurcation theory, it establishes the existence and orbital stability of nontrivial standing waves emerging at γ = 1, with stability depending on κ: stable for κ < 0, unstable for κ > 0, and critical behavior at κ = 0.

ABSTRACT

We consider a system of nonlinear Schrödinger equations related to the Raman amplification in a plasma. We study the orbital stability and instability of standing waves bifurcating from the semi-trivial standing wave of the system. The stability and instability of the semi-trivial standing wave at the bifurcation point are also studied. Moreover, we determine the set of the ground states completely.

Motivation & Objective

• To analyze the bifurcation structure of nontrivial standing wave solutions emerging from semi-trivial solutions in a two-component nonlinear Schrödinger system.
• To determine the orbital stability and instability of standing waves bifurcating from the semi-trivial state, particularly at the critical parameter value γ = 1.
• To completely characterize the set of ground states for the system using variational methods and symmetry arguments.
• To investigate the role of the parameter κ in determining stability when γ = 1, especially its sign dependence.

Proposed method

• Applies the Crandall-Rabinowitz local bifurcation theorem to identify γ = 1 as a bifurcation point for nontrivial solutions from the semi-trivial standing wave (0, e^{2iωt}φ_ω).
• Uses variational characterization of the ground state φ_ω as the minimizer of the ratio ‖v‖_{H^1_ω}^2 / ‖v‖_{L^3}^2 in H^1(ℝ^N)  \{0 }.
• Employs energy and charge functionals E(𝐮) and Q(𝐮) to define the constrained minimization problem for ground states in the space X = H^1(ℝ^N, ℂ)^2.
• Introduces group actions G(θ) and J to describe gauge and phase symmetries, and uses them to classify solutions via orbital equivalence.
• Applies Hölder and Gagliardo-Nirenberg inequalities to derive necessary conditions on L^3 norms of solutions, leading to the system of inequalities in (5.5).
• Uses the identity S_ω(𝐮) = d(ω) to relate the action functional to the ground state energy and derive the critical threshold ℓ = 6d(ω)/‖φ_ω‖_{L^3}^3.

Experimental results

Research questions

• RQ1 What is the structure of standing wave solutions bifurcating from the semi-trivial solution (0, e^{2iωt}φ_ω) at γ = 1?
• RQ2 How does the sign of the parameter κ affect the orbital stability of the bifurcating solutions when γ = 1?
• RQ3 What is the complete set of ground states for the system, and how are they characterized variationaly?
• RQ4 Under what conditions on (κ, γ) does the system admit nontrivial ground states beyond the semi-trivial state?
• RQ5 How do the L^3 norms of the components relate to the stability and bifurcation structure of the system?

Key findings

• For γ < 1, the semi-trivial standing wave (0, e^{2iωt}φ_ω) is orbitally stable; for γ > 1, it is unstable, independent of κ.
• At γ = 1, the stability of the semi-trivial solution depends on κ: stable if κ < 0, unstable if κ > 0, with critical behavior at κ = 0.
• When (κ, γ) ∈ 𝒦₁, the set of ground states ℒ_ω consists of solutions of the form (e^{iθ₁}α₊φ_ω(·+y₁), e^{iθ₂}β₋φ_ω(·+y₂)) with 2θ₁ − θ₂ ∈ 2πℤ and y₁ = y₂.
• When (κ, γ) ∈ 𝒦₂, the only ground states are of the form (0, e^{iθ}φ_ω(·+y)), corresponding to the semi-trivial state.
• When (κ, γ) ∈ 𝒦₃, the ground states are either the semi-trivial state or the bifurcated state from 𝒦₁, depending on parameter regime.
• The set of ground states is completely characterized: ℒ_ω = ℒ_ω¹ if (κ, γ) ∈ 𝒦₁, ℒ_ω = ℒ_ω⁰ if (κ, γ) ∈ 𝒦₂, and ℒ_ω = ℒ_ω⁰ ∪ ℒ_ω¹ if (κ, γ) ∈ 𝒦₃.