Instability of stationary solutions for double power nonlinear Schrödinger equations in one dimension

Noriyoshi Fukaya () and Masayuki Hayashi ()
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Noriyoshi Fukaya: Waseda University
Masayuki Hayashi: Kyoto University

Partial Differential Equations and Applications, 2025, vol. 6, issue 1, 1-23

Abstract: Abstract We consider a double power nonlinear Schrödinger equation possessing the algebraically decaying stationary solution $$\phi _0$$ ϕ 0 as well as exponentially decaying standing waves $$e^{i\omega t}\phi _\omega (x)$$ e i ω t ϕ ω ( x ) with $$\omega >0$$ ω > 0 . According to the general theory, stability properties of standing waves are determined by the derivative of $$\omega \mapsto M(\omega )\mathrel {\mathop :}=\frac{1}{2}\Vert \phi _\omega \Vert _{L^2}^2$$ ω ↦ M ( ω ) : = 1 2 ‖ ϕ ω ‖ L 2 2 ; namely $$e^{i\omega t}\phi _\omega $$ e i ω t ϕ ω with $$\omega >0$$ ω > 0 is stable if $$M'(\omega )>0$$ M ′ ( ω ) > 0 and unstable if $$M'(\omega )

Keywords: Nonlinear Schrödinger equation; Double power nonlinearities; Stationary solution; Instability; 35Q55; 35B35 (search for similar items in EconPapers)
Date: 2025
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DOI: 10.1007/s42985-024-00309-8

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