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In this paper, we present numerical methods for the determination of solitons, that consist in spatially localized stationary states of nonlinear scalar equations or coupled systems arising in nonlinear optics. We first use the well-known shooting method in order to find excited states (characterized by the number $k$ of nodes) for the classical nonlinear Schrödinger equation. Asymptotics can then be derived in the limits of either large $k$ are large nonlinear exponents $σ$ . In a second part, we compute...

Numerical computation of solitons for optical systems

Laurent Di Menza (2008)

ESAIM: Mathematical Modelling and Numerical Analysis

In this paper, we present numerical methods for the determination of solitons, that consist in spatially localized stationary states of nonlinear scalar equations or coupled systems arising in nonlinear optics. We first use the well-known shooting method in order to find excited states (characterized by the number k of nodes) for the classical nonlinear Schrödinger equation. Asymptotics can then be derived in the limits of either large k are large nonlinear exponents σ. In a second part, we compute...

On a q-deformed harmonic oscillator with variable linear momentum.

Harold Exton (1992)

Collectanea Mathematica

On an example of phase-space tunneling

Shu Nakamura (1995)

Annales de l'I.H.P. Physique théorique

On an inverse scattering problem for a discontinuous Sturm-Liouville equation with a spectral parameter in the boundary condition.

Mamedov, Khanlar R. (2010)

Boundary Value Problems [electronic only]

On Bojarski's index formula for nonsmooth interfaces.

Mitrea, Marius (1999)

Electronic Research Announcements of the American Mathematical Society [electronic only]

On nonlocal problems for ordinary differential equations and on a nonlocal parabolic transmission problem.

Denche, M. (1999)

Journal of Applied Mathematics and Stochastic Analysis

On the basis property of the root functions of differential operators with matrix coefficients

Oktay Veliev (2011)

Open Mathematics

We obtain asymptotic formulas for eigenvalues and eigenfunctions of the operator generated by a system of ordinary differential equations with summable coefficients and periodic or antiperiodic boundary conditions. Then using these asymptotic formulas, we find necessary and sufficient conditions on the coefficients for which the system of eigenfunctions and associated functions of the operator under consideration forms a Riesz basis.

On the differential operators with periodic matrix coefficients.

Veliev, O.A. (2009)

Abstract and Applied Analysis

On the essential spectra of matrix operators and applications.

Damak, Mondher, Jeribi, Aref (2007)

Electronic Journal of Differential Equations (EJDE) [electronic only]

On the essential spectrum of Dirac operators with spherically symmetric potentials.

Karl Michael Schmidt (1993)

Mathematische Annalen

On the genericity of nonvanishing instability intervals in periodic Dirac systems

Karl Michael Schmidt (1993)

Annales de l'I.H.P. Physique théorique

On the $L_{w}^{2}$ -solutions to general second-order nonsymmetric differential equations.

Sobhy El-sayed, I., Faried, N., Attia, G.M. (2001)

Siberian Mathematical Journal

On the Lyapunov exponent and exponential dichotomy for the quasi-periodic Schrödinger operator

R. Fabbri (2002)

Bollettino dell'Unione Matematica Italiana

In this paper we study the Lyapunov exponent $β (E)$ for the one-dimensional Schrödinger operator with a quasi-periodic potential. Let $Γ \subset R^{k}$ be the set of frequency vectors whose components are rationally independent. Let $Γ \subset R^{k}$ , and consider the complement in $Γ C^{r} (T^{k})$ of the set $D$ where exponential dichotomy holds. We show that $β = 0$ is generic in this complement. The methods and techniques used are based on the concepts of rotation number and exponential dichotomy.

On the singular spectrum for adiabatic quasiperiodic Schrödinger operators.

Marx, Magali, Najar, Hatem (2010)

Advances in Mathematical Physics

On the singular spectrum of discrete Schrödinger operator

S. Naboko (1993/1994)

Séminaire Équations aux dérivées partielles (Polytechnique)

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