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Title: Potential Well Theory for the Derivative Nonlinear Schrödinger Equation
Authors: HAYASHI, Masayuki
Author's alias: 林, 雅行
Keywords: 35Q55
35Q51
37K05
35A15
derivative nonlinear Schrödinger equation
solitons
potential well
variational methods
Issue Date: Oct-2020
Publisher: Research Institute for Mathematical Sciences, Kyoto University
Start page: 1
End page: 35
Thesis number: RIMS-1928
Abstract: We consider the following nonlinear Schrödinger equation of derivative type: (1) i∂tu + ∂2xu + i|u|2∂xu + b|u|4u = 0, (t, x) ∈ R×R, b ∈ R. If b = 0, this equation is known as a standard derivative nonlinear Schrödinger equation (DNLS), which is mass critical and completely integrable. The equation (1) can be considered as a generalized equation of DNLS while preserving mass criticality and Hamiltonian structure. For DNLS it is known that if the initial data u0∈H1(R) satisfies the mass condition ||u0||2L2 < 4π, the corresponding solution is global and bounded. In this paper we first establish the mass condition on (1) for general b∈R, which is exactly corresponding to 4π-mass condition for DNLS, and then characterize it from the viewpoint of potential well theory. We see that the mass threshold value gives the turning point in the structure of potential well generated by solitons. In particular, our results for DNLS give a characterization of both 4π-mass condition and algebraic solitons.
URI: http://hdl.handle.net/2433/261826
Related Link: http://www.kurims.kyoto-u.ac.jp/preprint/index.html
Appears in Collections:Research Institute for Mathematical Sciences, preprints

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