Soliton Solutions for the Lugiato–Lefever Equation by Analytical and Numerical Continuation Methods

Gärtner, Janina; Reichel, Wolfgang

doi:10.1007/978-3-030-47174-3_11

Cited by 2 publications

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References 23 publications

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“…Existence and bifurcation behavior of solutions of (1.1) have been studied quite well, cf. [8,9,12,13,21,23,24,25,26] and their stability properties have been investigated in [3,4,14,15,16,28,30]. Analytical and numerical investigations of (1.2) have recently been reported [10,11].…”

Section: Introductionmentioning

confidence: 99%

Pinning in the Extended Lugiato–Lefever Equation

Bengel,

Pelinovsky,

Reichel

2024

SIAM J. Math. Anal.

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We consider a variant of the Lugiato-Lefever equation (LLE), which is a nonlinear Schrödinger equation on a one-dimensional torus with forcing and damping, to which we add a first-order derivative term with a potential εV(x). The potential breaks the translation invariance of LLE. Depending on the existence of zeroes of the effective potential V eff , which is a suitably weighted and integrated version of V, we show that stationary solutions from ε = 0 can be continued locally into the range ε = 0. Moreover, the extremal points of the ε-continued solutions are located near zeros of V eff . We therefore call this phenomenon pinning of stationary solutions. If we assume additionally that the starting stationary solution at ε = 0 is spectrally stable with the simple zero eigenvalue due to translation invariance being the only eigenvalue on the imaginary axis, we can prove asymptotic stability or instability of its ε-continuation depending on the sign of V eff at the zero of V eff and the sign of ε. The variant of the LLE arises in the description of optical frequency combs in a Kerr nonlinear ring-shaped microresonator which is pumped by two different continuous monochromatic light sources of different frequencies and different powers. Our analytical findings are illustrated by numerical simulations.

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Section: Introductionmentioning

confidence: 99%

Pinning in the Extended Lugiato–Lefever Equation

Bengel,

Pelinovsky,

Reichel

2024

SIAM J. Math. Anal.

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Global continua of solutions to the Lugiato–Lefever model for frequency combs obtained by two-mode pumping

Elias

Jahnke

Kirn

et al. 2023

Z. Angew. Math. Phys.

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We consider Kerr frequency combs in a dual-pumped microresonator as time-periodic and spatially $$2\pi $$ 2 π -periodic traveling wave solutions of a variant of the Lugiato–Lefever equation, which is a damped, detuned and driven nonlinear Schrödinger equation given by $$\textrm{i}a_\tau =(\zeta -\textrm{i})a-d a_{x x}-|a|^2a+\textrm{i}f_0+\textrm{i}f_1\textrm{e}^{\textrm{i}(k_1 x-\nu _1 \tau )}$$ i a τ = ( ζ - i ) a - d a xx - | a | 2 a + i f 0 + i f 1 e i ( k 1 x - ν 1 τ ) . The main new feature of the problem is the specific form of the source term $$f_0+f_1\textrm{e}^{\textrm{i}(k_1 x-\nu _1 \tau )}$$ f 0 + f 1 e i ( k 1 x - ν 1 τ ) which describes the simultaneous pumping of two different modes with mode indices $$k_0=0$$ k 0 = 0 and $$k_1\in \mathbb {N}$$ k 1 ∈ N . We prove existence and uniqueness theorems for these traveling waves based on a-priori bounds and fixed point theorems. Moreover, by using the implicit function theorem and bifurcation theory, we show how non-degenerate solutions from the 1-mode case, i.e., $$f_1=0$$ f 1 = 0 , can be continued into the range $$f_1\not =0$$ f 1 ≠ 0 . Our analytical findings apply both for anomalous ($$d>0$$ d > 0 ) and normal ($$d<0$$ d < 0 ) dispersion, and they are illustrated by numerical simulations.

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Soliton Solutions for the Lugiato–Lefever Equation by Analytical and Numerical Continuation Methods

Cited by 2 publications

References 23 publications

Pinning in the Extended Lugiato–Lefever Equation

Pinning in the Extended Lugiato–Lefever Equation

Global continua of solutions to the Lugiato–Lefever model for frequency combs obtained by two-mode pumping

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