Sharp thresholds for stability and instability of standing waves in a double power nonlinear Schrödinger equation

Hayashi, Masayuki

doi:10.48550/arxiv.2112.07540

Cited by 2 publications

(2 citation statements)

References 8 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…After the first version of this paper was posted to arXiv, Professor Hayashi kindly informed us he had an independent similar result and posted it as [22]. His Theorem 1.3 is similar to our Theorem 1.1 although it does not include the case 1 < p < 9/5.…”

Section: Introductionmentioning

confidence: 92%

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

Kfoury¹,

Coz²,

Tsai³

2021

Preprint

View full text Add to dashboard Cite

For the double power one dimensional nonlinear Schrödinger equation, we establish a complete classification of the stability or instability of standing waves with positive frequencies. In particular, we fill out the gaps left open by previous studies. Stability or instability follows from the analysis of the slope criterion of Grillakis, Shatah and Strauss. The main new ingredients in our approach are a reformulation of the slope and the explicit calculation of the slope value in the zero-frequency case. Our theoretical results are complemented with numerical experiments.

show abstract

Section: Introductionmentioning

confidence: 92%

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

Kfoury¹,

Coz²,

Tsai³

2021

Preprint

View full text Add to dashboard Cite

show abstract

“…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

View full text Add to dashboard Cite

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

show abstract

Sharp thresholds for stability and instability of standing waves in a double power nonlinear Schrödinger equation

Cited by 2 publications

References 8 publications

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

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

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

Contact Info

Product

Resources

About