Asymptotics for the higher-order derivative nonlinear Schrödinger equation

Pavel I. Naumkin; Isahi Sánchez-Suárez

doi:10.3934/cpaa.2021028

Article Contents

2021, Volume 20, Issue 4: 1447-1478. Doi: 10.3934/cpaa.2021028

This issue Previous Article A Hopf type lemma and the symmetry of solutions for a class of Kirchhoff equations Next Article On the uniqueness of solutions of a semilinear equation in an annulus

Asymptotics for the higher-order derivative nonlinear Schrödinger equation

Pavel I. Naumkin^1, and
Isahi Sánchez-Suárez^2, ,

1.
Centro de Ciencias Matemáticas, UNAM Campus Morelia, AP 61-3 (Xangari), Morelia CP 58089, Michoacán, MEXICO
2.
Universidad Politécnica de Uruapan, CP 60210 Uruapan Michoacán, MEXICO

^* Corresponding author
^* Corresponding author

Received: August 31, 2020

Revised: December 31, 2020

Early access: March 2021

Published: April 2021

The work of P. I. N. is partially supported by CONACYT project 283698 and PAPIIT project IN103221

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Abstract

We study the Cauchy problem for the derivative higher-order nonlinear Schrödinger equation

$ \begin{cases} i\partial_{t}v+\dfrac{a}{2}\partial_{x}^{2}v-\dfrac{b}{4}\partial_{x} ^{4}v = \left( \overline{\partial_{x}v}\right) ^{2},\text{ }t>1,\text{ } x\in\mathbb{R},\\ v\left( 1,x\right) = v_{0}\left( x\right) ,\text{ }x\in\mathbb{R}\text{,} \end{cases} $

where $ a,b>0. $ Our aim is to prove global existence and calculate the large time asymptotics of solutions. We develop the factorization techniques originated in papers [13,10,12]. Also we follow the method of papers [9,11] to transform the quadratic nonlinearity to critical cubic nonlinearities similarly to the normal forms of Shatah [18].

Keywords:

Asymptotics of solutions,
derivative nonlinear Schrödinger equation,
higher order,
factorization techniques,
modified scattering.

Mathematics Subject Classification: Primary: 35B40; Secondary: 35Q35.

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References

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