Stability of a stationary solution for the Lugiato-Lefever equation

Miyaji, Tomoyuki; Ohnishi, Isamu

doi:10.2748/tmj/1325886285

Cited by 11 publications

(12 citation statements)

References 15 publications

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“…More precisely, these solutions are nonlinearly stable if α < 41/30 and nonlinearly unstable if α > 41/30 (see [9]). Moreover, it is proved in [10], using a Strichartz estimate, that this family is orbitaly stable with respect to L 2 perturbations for α < 41/30.…”

Section: Discussionmentioning

confidence: 98%

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A bifurcation analysis for the Lugiato-Lefever equation

Godey

2017

Eur. Phys. J. D

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The Lugiato-Lefever equation is a cubic nonlinear Schrödinger equation, including damping, detuning and driving, which arises as a model in nonlinear optics. We study the existence of stationary waves which are found as solutions of a four-dimensional reversible dynamical system in which the evolutionary variable is the space variable. Relying upon tools from bifurcation theory and normal forms theory, we discuss the codimension 1 bifurcations. We prove the existence of various types of steady solutions, including spatially localized, periodic, or quasiperiodic solutions.

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Section: Discussionmentioning

confidence: 98%

“…The existence of 2π−periodic stationary solutions satisfying Neumann boundary conditions has been proved in [8], provided that the coefficients α and F belong to suitable ranges. Moreover, the stability of a family of periodic solutions has been discussed in [10].…”

Section: Introductionmentioning

confidence: 99%

A bifurcation analysis for the Lugiato-Lefever equation

Godey

2017

Eur. Phys. J. D

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“…As explained in Remark ( 10) we can ignore the turning points whereζ = 0. Hence, inserting (21) into the transversality condition (20) we get in case Im(a 2 0 ) ≤ 0…”

Section: Proof Of (I)mentioning

confidence: 99%

“…Convergence results for the numerical Strang-splitting scheme can be found in [11]. Finally, the orbital asymptotic stability of 2π-periodic solutions was investigated in [29] (Theorem 1) with the aid of the Gearhart-Prüss-Theorem, see also [18,20]. Notice that the linearized operators (i.e.…”

Section: Introductionmentioning

confidence: 99%

The Lugiato–Lefever Equation with Nonlinear Damping Caused by Two Photon Absorption

2021

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In this paper we investigate the effect of nonlinear damping on the Lugiato-Lefever equation|a| 2 a + if on the torus or the real line. For the case of the torus it is shown that for small nonlinear damping κ > 0 stationary spatially periodic solutions exist on branches that bifurcate from constant solutions whereas all nonconstant solutions disappear when the damping parameter κ exceeds a critical value. These results apply both for normal (d < 0) and anomalous (d > 0) dispersion. For the case of the real line we show by the Implicit Function Theorem that for small nonlinear damping κ > 0 and large detuning ζ 1 and large forcing f 1 strongly localized, bright solitary stationary solutions exists in the case of anomalous dispersion d > 0. These results are achieved by using techniques from bifurcation and continuation theory and by proving a convergence result for solutions of the time-dependent Lugiato-Lefever equation.

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“…In the context of mathematics, the well-posedness and the existence of the global attractor for a damped-driven nonlinear Schrödinger equation have been studied on some regions: a finite interval [7,19], open bounded subset of R 2 [1], on a unit disk of R 2 [17], and R N with N ≤ 3 [10]. Steady-state bifurcation for LLE on T 1 and R 1 [12] and the stability of bifurcating solution in L 2 (T 1 ) [13] have been studied by the present authors.…”

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confidence: 99%

Steady-state mode interactions of radially symmetric modes for the Lugiato-Lefever equation on a disk

Miyaji¹

2018

Communications on Pure &Amp; Applied Analysis

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We study a nonlinear Schrödinger equation with damping, detuning, and spatially homogeneous input terms, which is called the Lugiato-Lefever equation, on the unit disk with the Neumann boundary conditions. We aim at understanding bifurcations of a so-called cavity soliton which is a radially symmetric stationary spot solution. It is known by numerical simulations that a cavity soliton bifurcates from a spatially homogeneous steady state. We prove the existence of the parameter-dependent center manifold and a branch of radially symmetric steady state in a neighborhood of the bifurcation point. In order to capture further bifurcations of the radially symmetric steady state, we study a degenerate bifurcation for which two radially symmetric modes become unstable simultaneously, which is called the two-mode interaction. We derive a vector field on the center manifold in a neighborhood of such a degenerate bifurcation and present numerical simulations to demonstrate the Hopf and homoclinic bifurcations of bifurcating solutions.2000 Mathematics Subject Classification. Primary: 35Q55; Secondary: 35B36.

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Stability of a stationary solution for the Lugiato-Lefever equation

Cited by 11 publications

References 15 publications

A bifurcation analysis for the Lugiato-Lefever equation

A bifurcation analysis for the Lugiato-Lefever equation

The Lugiato–Lefever Equation with Nonlinear Damping Caused by Two Photon Absorption

Steady-state mode interactions of radially symmetric modes for the Lugiato-Lefever equation on a disk

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