Existence and stability of TE modes in a stratified non-linear dielectric

Stuart, Charles Alexander

doi:10.1093/imamat/hxm033

Cited by 8 publications

(3 citation statements)

References 16 publications

(24 reference statements)

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“…Self-focusing planar waveguides have been thoroughly investigated, both from the physical and the mathematical standpoints. Amongst the wide literature about nonlinear waveguides, let us mention [1,3,19,21,31,33] regarding the physics, and [8,20,22,25,27,29] for some mathematical results. Historically, the mathematical study of (NLS) has grown in parallel to -and was largely motivated by -the development of nonlinear waveguide theory.…”

Section: Self-focusing Planar Waveguidesmentioning

confidence: 99%

“…As explained in [8, Section 6.3], a global bifurcation result such as Theorem 2.10 enables one to make the paraxial approximation as accurate as desired. Hence stability of the travelling waves (4.3) can be inferred from Theorem 3.8 -the physical meaning of 'stability' for the TE travelling waves (4.3) is discussed in Section 6.2 of [25], where Kerr media are studied using the results in [15,16,24]. The rigorous proof of stability of the travelling waves (4.3), thus obtained from the analysis of (NLS), puts on firm mathematical grounds the phenomenon of selftrapping, known to physicists since the early 70's (first predicted theoretically and later observed experimentally -see [21]).…”

Section: Self-focusing Planar Waveguidesmentioning

confidence: 99%

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Orbitally stable standing waves for the asymptotically linear one-dimensional NLS

Genoud¹

2013

Evolution Equations &Amp; Control Theory

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In this article we study the one-dimensional, asymptotically linear, non-linear Schrödinger equation (NLS). We show the existence of a global smooth curve of standing waves for this problem, and we prove that these standing waves are orbitally stable. As far as we know, this is the first rigorous stability result for the asymptotically linear NLS. We also discuss an application of our results to self-focusing waveguides with a saturable refrac-1

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Section: Self-focusing Planar Waveguidesmentioning

confidence: 99%

Section: Self-focusing Planar Waveguidesmentioning

confidence: 99%

Orbitally stable standing waves for the asymptotically linear one-dimensional NLS

Genoud¹

2013

Evolution Equations &Amp; Control Theory

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“…One example is in the study of guided waves propagating through a non-linear optical material. In this context, a derivation of (CNLS) from first principles and an interpretation of standing waves and their orbital stability is given in [31]. See equation (7.9) of [31] for (CNLS) and then Section 7.2 of the same paper for a situation leading to the hypotheses introduced in Section 8.3 of these notes.…”

Section: The Nonlinear Schrödinger Equationmentioning

confidence: 99%

Lectures on the Orbital Stability of Standing Waves and Application to the Nonlinear Schrödinger Equation

Stuart

2008

Milan j. math.

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In the first part of these notes, we deal with first order Hamiltonian systems in the form Ju (t) = ∇H(u(t)) where the phase space X may be infinite dimensional so as to accommodate some partial differential equations. The Hamiltonian H ∈ C 1 (X, R) is required to be invariant with respect to the action of a group {e tA : t ∈ R} of isometries where A ∈ B(X, X) is skew-symmetric and JA = AJ. A standing wave is a solution having the form u(t) = e tλA ϕ for some λ ∈ R and ϕ ∈ X such that λJAϕ = ∇H(ϕ). Given a solution of this type, it is natural to investigate its stability with respect to perturbations of the initial condition. In this context, the appropriate notion of stability is orbital stability in the usual sense for a dynamical system. We present some of the important criteria for establishing orbital stability of standing waves.In the second part we consider the nonlinear Schrödinger equation which provides an interesting example of this situation where standing waves appear as time-harmonic solutions. We show how the general theory applies to this case and review what is known about stability. Mathematics Subject Classification (2000). Primary 37-01; Secondary 37K45, 35Q55.

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A Concentration Phenomenon for Semilinear Elliptic Equations

Ackermann

Szulkin

2012

Arch Rational Mech Anal

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For a domain Ω ⊂ R N we consider the equation −∆u+V (x)u = Q n (x)|u| p−2 u with zero Dirichlet boundary conditions and p ∈ (2, 2 * ). Here V ≥ 0 and Q n are bounded functions that are positive in a region contained in Ω and negative outside, and such that the sets {Q n > 0} shrink to a point x 0 ∈ Ω as n → ∞. We show that if u n is a nontrivial solution corresponding to Q n , then the sequence (u n ) concentrates at x 0 with respect to the H 1 and certain L q -norms. We also show that if the sets {Q n > 0} shrink to two points and u n are ground state solutions, then they concentrate at one of these points.

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Existence and stability of TE modes in a stratified non-linear dielectric

Cited by 8 publications

References 16 publications

Orbitally stable standing waves for the asymptotically linear one-dimensional NLS

Orbitally stable standing waves for the asymptotically linear one-dimensional NLS

Lectures on the Orbital Stability of Standing Waves and Application to the Nonlinear Schrödinger Equation

A Concentration Phenomenon for Semilinear Elliptic Equations

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