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Local theory and long time behaviour of solutions of nonlinear Schrodinger equations

Abstract : Our main research is to study nonlinear Schrodinger equations, especially derivative nonlinear Schrödinger equations. Our first goal is to answer questions about the existence and uniqueness of solutions, considering whether the time of existence is finite or infinite and checking whether solutions depend continuously on initial data or not. When solutions exist, we study the behaviour of solutions at large times, answering questions about stability and instability of solitons/algebraic standing waves/periodic waves, the existence of blow-up solutions, the existence of multi-soliton trains, and the existence of multi kink-soliton trains. The Cauchy problem of the derivative nonlinear Schrodinger equations was treated many in the Sobolev space H^1(R). The first main goal is to establish local theory with nonvanishing boundary conditions. We use a Gauge transform method to transfer the original equation into a system without derivative terms. By studying the Cauchy problem of this system, we obtain the results for the original equation. Next, we consider the nonlinear Schrodinger equation with derivative nonlinearities in half-line case with Robin boundary condition. We prove the existence of blow-up solutions. Moreover, we prove that the equation admits the special solutions which are called standing waves. This solution is a minimizer of a variational problem. We prove the stability and instability of this kind of solution depending on the sign of the given Robin condition parameter. Next, we investigate the multi solitons theory of derivative nonlinear Schrodinger equations. The existence of this kind of solution shows that there exists a global solution with an arbitrary large of initial data. The method used for classical nonlinear Schrodinger equations can not apply in this case with derivative nonlinearities. We take advantage of Gauge transform to overcome this difficulty. Finally, we consider the nonlinear Schrodinger equation with triple power. Our goal is to prove the instability of algebraic standing waves i.e the standing waves with zero frequency. Our motivation is from the work of Liu-Tsai-Zwiers where the authors gave a picture of the stability and instability of standing waves of triple power NLS in one dimension and the work of Fukaya-Hayashi where the authors proved the instability of algebraic waves in the case of double power NLS.
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Submitted on : Thursday, August 25, 2022 - 3:42:11 PM
Last modification on : Friday, August 26, 2022 - 3:58:20 AM

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  • HAL Id : tel-03760769, version 1

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van Tin Phan. Local theory and long time behaviour of solutions of nonlinear Schrodinger equations. Analysis of PDEs [math.AP]. Université Paul Sabatier - Toulouse III, 2022. English. ⟨NNT : 2022TOU30077⟩. ⟨tel-03760769⟩

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