Analysis of stability and instability for standing waves of the double power one dimensional nonlinear Schr{ö}dinger equation

Kfoury, Perla; Coz, Stefan Le; Tsai, Tai‐Peng

doi:10.48550/arxiv.2112.06529

Cited by 1 publication

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“…Several studies have been made on the asymptotic behavior of solutions to double power nonlinear Schrödinger equations (see e.g. [3,6,11,19,20,22,23,26,27,28,30,32,33,39,40,41] and references therein). Here, we are concerned with global dynamics of solutions whose mass and energy equal to those of the ground state.…”

Section: Introductionmentioning

confidence: 99%

Threshold solutions for the 3D focusing cubic-quintic nonlinear Schrodinger equation at low frequencies

Hamano¹,

Kikuchi²,

Watanabe³

2022

Preprint

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This paper addresses the focusing cubic-quintic nonlinear Schrödinger equation in three space dimensions. Especially, we study the global dynamics of solutions whose energy and mass equal to those of the ground state in the sprits of Duyckaerts and Merle [14]. When we try to obtain the corresponding results of [14], we meet several difficulties due to the cubic-quintic nonlinearity. We overcome them by using the one-pass theorem (no return theorem) developed by Nakanishi and Schlag [38].In this paper, we address the threshold solution to (1.1). To state our results, we putClearly, we have BA ω = BA ω,− ∪ BA ω,0 ∪ BA ω,+ . We see from Proposition 2.1 below that BA ω,0 = e iθ Q ω : θ ∈ R .(1.10)In addition, we can easily find that the sets BA ω,± and BA ω,0 are invariant under the flow of (1.1) (see e.g. Lemma 4.2 below). Then, by a similar argument to [14], we can construct the following special solutions to (1.1):Theorem 1.2. There exists a sufficiently small ω * > 0 such that for ω ∈ (0, ω * ), (1.1) has two radial solutions Q + ω ∈ BA ω,+ and Q − ω ∈ BA ω,− satisfying the following:(i) Q ± ω exists on [0, ∞), and there exist constants e ω , C ω > 0 such that

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