A uniqueness result for the Sine-Gordon breather

Mandel, Rainer

doi:10.1007/s42985-021-00084-w

Cited by 2 publications

(1 citation statement)

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“…Time-periodic solutions with finite energy are called breather solutions. However, it cannot be expected that such solutions with finite energy do exist in general, according to the non-persistence of breathers result for nonlinear wave equations in homogeneous media [Den93,BMW94,Man21]. Nevertheless, generalised breather solutions, i.e.…”

Section: Introductionmentioning

confidence: 99%

Travelling modulating pulse solutions with small tails for a nonlinear wave equation in periodic media

Dohnal,

Pelinovsky,

Schneider

2024

Nonlinearity

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Travelling modulating pulse solutions consist of a small amplitude pulse-like envelope moving with a constant speed and modulating a harmonic carrier wave. Such solutions can be approximated by solitons of an effective nonlinear Schrödinger equation arising as the envelope equation. We are interested in a rigorous existence proof of such solutions for a nonlinear wave equation with spatially periodic coefficients. Such solutions are quasi-periodic in a reference frame co-moving with the envelope. We use spatial dynamics, invariant manifolds, and near-identity transformations to construct such solutions on large domains in time and space. Although the spectrum of the linearised equations in the spatial dynamics formulation contains infinitely many eigenvalues on the imaginary axis or in the worst case the complete imaginary axis, a small denominator problem is avoided when the solutions are localised on a finite spatial domain with small tails in far fields.

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Section: Introductionmentioning

confidence: 99%