Construction of breather solutions for nonlinear Klein-Gordon equations on periodic metric graphs

Maier, Daniela

doi:10.1016/j.jde.2019.09.035

Cited by 5 publications

(8 citation statements)

References 39 publications

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“…Our results depend strongly on the assumption that (at least some of) the coefficients s(x), q(x), V (x) in (1 ± ) and (2) depend on x. In this sense we are close to the results in [12] and [13,16] since these papers also make strong use of spatially varying coefficients. However, even more important in our setting is the particular property of the curl-operator to annihilate gradient fields.…”

Section: Introductionsupporting

confidence: 75%

“…For semilinear scalar 1+1-dimensional wave equations with non-constant coefficients it is a challenging task to find bright breathers. This was accomplished for the first time in [12] by making use of spatial dynamics and center manifold reduction, and subsequently by variational methods in [13][14][15][16]. While the physics literature on rogue waves is quite abundant, cf.…”

Section: Introductionmentioning

confidence: 99%

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Breathers and rogue waves for semilinear curl-curl wave equations

Plum

Reichel

2023

J Elliptic Parabol Equ

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We consider localized solutions of variants of the semilinear curl-curl wave equation $$s(x) \partial _t^2 U +\nabla \times \nabla \times U + q(x) U \pm V(x) \vert U \vert ^{p-1} U = 0$$ s ( x ) ∂ t 2 U + ∇ × ∇ × U + q ( x ) U ± V ( x ) | U | p - 1 U = 0 for $$(x,t)\in {\mathbb {R}}^3\times {\mathbb {R}}$$ ( x , t ) ∈ R 3 × R and arbitrary $$p>1$$ p > 1 . Depending on the coefficients s, q, V we can prove the existence of three types of localized solutions: time-periodic solutions decaying to 0 at spatial infinity, time-periodic solutions tending to a nontrivial profile at spatial infinity (both types are called breathers), and rogue waves which converge to 0 both at spatial and temporal infinity. Our solutions are weak solutions and take the form of gradient fields. Thus they belong to the kernel of the curl-operator so that due to the structural assumptions on the coefficients the semilinear wave equation is reduced to an ODE. Since the space dependence in the ODE is just a parametric dependence we can analyze the ODE by phase plane techniques and thus establish the existence of the localized waves described above. Noteworthy side effects of our analysis are the existence of compact support breathers and the fact that one localized wave solution U(x, t) already generates a full continuum of phase-shifted solutions $$U(x,t+b(x))$$ U ( x , t + b ( x ) ) where the continuous function $$b:{\mathbb {R}}^3\rightarrow {\mathbb {R}}$$ b : R 3 → R belongs to a suitable admissible family.

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Section: Introductionsupporting

confidence: 75%

Section: Introductionmentioning

confidence: 99%

Breathers and rogue waves for semilinear curl-curl wave equations

Plum

Reichel

2023

J Elliptic Parabol Equ

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“…Our existence result in Theorem 1.1 for spatially localized, real-valued time-periodic solutions of (1) is the main outcome of this paper, and it directly compares to the result from [16]. Methodically, we also use tools from the calculus of variations like in [10].…”

Section: Introduction and Resultsmentioning

confidence: 89%

“…Now the heterogeneity stems from the underlying branched structure and not from the equation or the operator. While at first [21,22] standing monochromatic waves were of interest for NLS equations on quantum graphs, more recently Maier [16] gave the first existence proof of real-valued breathers for (1). Her method first produced spectral gaps in the wave operator by the correct choice of the temporal frequency.…”

Section: Introduction and Resultsmentioning

confidence: 99%

Breather solutions for a semilinear Klein-Gordon equation on a periodic metric graph

Maier

Reichel

Schneider

2023

Journal of Mathematical Analysis and Applications

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“…In [BCLS11] breather solutions were constructed by spatial dynamics in the phase space of time-periodic solutions, invariant manifold theory and normal form theory. With the same approach in [Mai20] such solutions were constructed for a cubic Klein-Gordon equation on an infinite periodic necklace graph. The existence of large amplitude breather solutions of the semi-linear wave equation (1) was shown in [HR19,MS21] via a variational approach.…”

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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Construction of breather solutions for nonlinear Klein-Gordon equations on periodic metric graphs

Cited by 5 publications

References 39 publications

Breathers and rogue waves for semilinear curl-curl wave equations

Breathers and rogue waves for semilinear curl-curl wave equations

Breather solutions for a semilinear Klein-Gordon equation on a periodic metric graph

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

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