Initial boundary value problems for some nonlinear dispersive models on the half-line: a review and open problems

Cavalcante, Márcio

doi:10.1007/s40863-019-00144-5

Cited by 5 publications

(2 citation statements)

References 39 publications

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“…For a complete survey about the problems (3.1) and (3.2) the reader can see [16]. Also, results of orbital stability and asymptotic stability can be seen in [17] and [18].…”

Section: Duhamel Operatormentioning

confidence: 99%

Linear instability criterion for the Korteweg–de Vries equation on metric star graphs

Pava

Cavalcante

2021

Nonlinearity

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The aim of this work is to establish a novel linear instability criterion for the Korteweg–de Vries (KdV) model on metric graphs. In the case of balanced graphs with a structure represented by a finite collection of semi-infinite edges and with boundary condition of δ-type interaction at the graph-vertex, we show that the continuous tail and bump profiles are linearly unstable. In this case, the use of the analytic perturbation theory of operators as well as the extension theory of symmetric operators is fundamental in our stability analysis. The arguments showed in this investigation have prospects in the study of the instability of stationary waves solutions for nonlinear evolution equations on metric graph.

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“…For a complete survey about the problems (3.1) and (3.2) the reader can see [16]. Also, results of orbital stability and asymptotic stability can be seen in [17] and [18].…”

Section: Duhamel Operatormentioning

confidence: 99%

Linear instability criterion for the Korteweg–de Vries equation on metric star graphs

Pava

Cavalcante

2021

Nonlinearity

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“…In the last years the study of initial boundary value problems for nonlinear dispersive models on the half-lines has given attention of many researchers (see [7] for a survey about the topic.) This turns out to be a rather challenging problem, since the more natural techniques used in standard domains, as R n or T does not work directly in the context of the half-lines due the lack of Fourier transform.…”

Section: Introductionmentioning

confidence: 99%

The nonlinear Quadratic Interactions of the Schrödinger type on the half-line

Barbosa¹,

Cavalcante²

2021

Preprint

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In this work we study the initial boundary value problem associated with the coupled Schrödinger equations with quadratic nonlinearities, that appears in nonlinear optics, on the half-line. We obtain local well-posedness for data in Sobolev spaces with low regularity, by using a forcing problem on the full line with a presence of a forcing term in order to apply the Fourier restriction method of Bourgain. The crucial point in this work is the new bilinear estimates on the classical Bourgain spaces X s,b with b < 1 2 , jointly with bilinear estimates in adapted Bourgain spaces that will used to treat the traces of nonlinear part of the solution. Here the understanding of the dispersion relation is the key point in these estimates, where the set of regularity depends strongly of the constant a measures the scaling-diffraction magnitude indices.

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Sharp well-posedness of the biharmonic Schrödinger equation in a quarter plane

Compaan,

Tzirakis

2023

Partial Differ. Equ. Appl.

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Initial boundary value problems for some nonlinear dispersive models on the half-line: a review and open problems

Cited by 5 publications

References 39 publications

Linear instability criterion for the Korteweg–de Vries equation on metric star graphs

Linear instability criterion for the Korteweg–de Vries equation on metric star graphs

The nonlinear Quadratic Interactions of the Schrödinger type on the half-line

Sharp well-posedness of the biharmonic Schrödinger equation in a quarter plane

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