Introduction to Nonlinear Dispersive Equations

Linares, Felipe; Ponce, Gustavo

doi:10.1007/978-1-4939-2181-2

Cited by 206 publications

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“…Such estimates will be then used in Section 3 to perform a fixed point argument and to show the local well-posedness for the Cauchy problems (1.6) and (1.7). By using the conservation of mass in the case of a L 2 −subcritical power-type nonlinearity we also prove the global well-posedness (see the seminal paper by Tsutsumi [11], and also the monographs [1], [10], [8]), by obtaining some uniform bounds for {u ω } in ω. In Section 4 we prove the main result of this paper, Theorem 1.1.…”

Section: Introduction and Statement Of The Main Resultsmentioning

confidence: 91%

On the XFEL Schrödinger Equation: Highly Oscillatory Magnetic Potentials and Time Averaging

Antonelli

Athanassoulis

Hajaiej

et al. 2014

Arch Rational Mech Anal

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Abstract. We analyse a nonlinear Schrödinger equation for the time-evolution of the wave function of an electron beam, interacting selfconsistently through a Hartree-Fock nonlinearity and through the repulsive Coulomb interaction of an atomic nucleus. The electrons are supposed to move under the action of a time dependent, rapidly periodically oscillating electromagnetic potential. This can be considered a simplified effective single particle model for an X-ray Free Electron Laser (XFEL). We prove the existence and uniqueness for the Cauchy problem and the convergence of wave-functions to corresponding solutions of a Schrödinger equation with a time-averaged Coulomb potential in the high frequency limit for the oscillations of the electromagnetic potential.

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Section: Introduction and Statement Of The Main Resultsmentioning

confidence: 91%

On the XFEL Schrödinger Equation: Highly Oscillatory Magnetic Potentials and Time Averaging

Antonelli

Athanassoulis

Hajaiej

et al. 2014

Arch Rational Mech Anal

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“…where we have suppressed constants depending on l. Integrating in the time interval [0, t], we see 17) as factors in the summation are estimated via (2.11) and the inductive hypothesis.…”

Section: Proof Of Theoremmentioning

confidence: 99%

Propagation of Regularity and Persistence of Decay for Fifth Order Dispersive Models

Segata

Smith

2015

J Dyn Diff Equat

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Abstract. This paper considers the initial value problem for a class of fifth order dispersive models containing the fifth order KdV equationThe main results show that regularity or polynomial decay of the data on the positive half-line yields regularity in the solution for positive times. IntroductionIn this work we study propagation of regularity and persistence of decay results for a class of fifth order dispersive models. For concreteness, the main theorems are stated for initial value problems of the formwhere c j are real constants, u : R × R → R is an unknown function and u 0 : R → R is a given function. Eq. (1.1) contains the specific equationwhich is the third equation in the sequence of nonlinear dispersive equationsknown as the KdV hierarchy. Here the polynomials Q j are chosen so that equation (1.3) has the Lax pair formulationThe first two equations in the hierarchy areand the KdV equationWith only slight modifications concerning the hypothesis on the initial data, the techniques in this paper apply to a large class of fifth order equations including the following models arising from mathematical physics:

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“…In this section we provide some preliminary results that we shall use later. Let us recall two important lemmas whose proofs can be found in [1].…”

Section: Preliminariesmentioning

confidence: 99%

A NOTE ON THE EXISTENCE OF SOLITARY WAVES TO THE REGULARIZED BENJAMIN-ONO ZAKHAROV-KUZNETSOV (rBO-ZK) EQUATION

Salazar¹,

Higuera²

2015

Int. J. of Appl. Math.

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Introduction to Nonlinear Dispersive Equations

Cited by 206 publications

References 0 publications

On the XFEL Schrödinger Equation: Highly Oscillatory Magnetic Potentials and Time Averaging

On the XFEL Schrödinger Equation: Highly Oscillatory Magnetic Potentials and Time Averaging

Propagation of Regularity and Persistence of Decay for Fifth Order Dispersive Models

A NOTE ON THE EXISTENCE OF SOLITARY WAVES TO THE REGULARIZED BENJAMIN-ONO ZAKHAROV-KUZNETSOV (rBO-ZK) EQUATION

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