Traveling waves in Fermi-Pasta-Ulam lattices with saturable nonlinearities

Панков, А. А.; Rothos, Vassilios M.

doi:10.3934/dcds.2011.30.835

Cited by 25 publications

(19 citation statements)

References 15 publications

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“…In the case of FPU chains with saturable nonlinearities [20], for example, |x| p ) is bounded, the conclusion of Theorem 6.3 holds true if the nonresonance condition is satisfied. Other examples can be found in [20].…”

Section: Asymptotic Quadratic Potentialmentioning

confidence: 66%

“…The purpose of emphasizing such a conclusion for monatomic chains is to compare our results with Pankov's conclusion (see Proposition 4.4 in [19]) that the supersonic (in our notation ω/k > √ K) periodic traveling waves of form (in our notation) Similarly, for asymptotic quadratic potential W (x) = Kx 2 /2+U (x), if xU (x) < 0 for all x = 0, then the wave speed of traveling waves of form (7.2) has an upper bound and nontrivial wave trains of form (7.2) do not exist if the wave speed does not belong to the allowed interval; see Proposition 1 of section 6 in [20].…”

Section: Asymptotic Quadratic Potentialmentioning

confidence: 98%

“…Application to monatomic chains. The existence of wave trains in monatomic FPU chains has been extensively studied; see [5,7,10,13,14,18,19,20,28] and the references therein. The aim of this section is to apply the conclusions of Theorems A and B to monatomic chains to obtain some new results.…”

Section: Asymptotic Quadratic Potentialmentioning

confidence: 99%

“…Herrmann [13] has proven the existence of unimodal traveling waves under similar assumptions. Pankov and Rothos [20] investigated the existence of traveling waves for FPU chains with saturable nonlinearities under some nonresonance condition. Following the idea in [21], in which the periodic traveling waves for the monatomic Frenkel-Kontorova model were studied, we apply the saddle point theorem to discuss the existence of wave trains with two different waveform functions for diatomic FPU chains with asymptotic quadratic potential.…”

Section: Introductionmentioning

confidence: 99%

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Wave Propagation in Diatomic Lattices

Qin¹

2015

SIAM J. Math. Anal.

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We study periodic traveling waves (wave trains) in diatomic Fermi-Pasta-Ulam chains (FPU). By applying the minimax principle, we demonstrate the existence of two different periodic waveform functions corresponding, respectively, to light and heavy particles. Our approach applies to the FPU β-model for each wavenumber and each frequency, and to FPU chains with asymptotic quadratic potential for wavenumbers and frequencies satisfying the nonresonance condition. As an application to monatomic lattices, we show for the monatomic soft FPU β-model the existence of supersonic wave trains with two different waveform functions for adjacent particles, contrary to the nonexistence of supersonic wave trains with only one waveform function.

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Section: Asymptotic Quadratic Potentialmentioning

confidence: 66%

Section: Asymptotic Quadratic Potentialmentioning

confidence: 98%

Section: Asymptotic Quadratic Potentialmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Wave Propagation in Diatomic Lattices

Qin¹

2015

SIAM J. Math. Anal.

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“…Bak [1] obtained the existence of periodic traveling waves, which extends the previous theorem to a wide range of speed. In [13], Pankov and Rothos discussed a similar problem with saturable nonlinearities by the same method.…”

Section: Introduction and Main Resultsmentioning

confidence: 93%

Periodic and Solitary Travelling Waves in Infinite Lattices without Ambrosetti–Rabinowitz Condition

Zhang

2014

J Dyn Diff Equat

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In this paper, we consider FPU lattices with particles of unit mass. The dynamics of the system is described by the infinite system of second order differential equationswhere q n denotes the displacement of the n-th lattice site and U is the potential of interaction between two adjacent particles. We investigate the existence of two kinds travelling wave solutions: periodic and solitary ones under some growth conditions on U which is different from the widely used Ambrosetti-Rabinowitz condition.

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Periodic traveling wave solutions of discrete nonlinear Klein‐Gordon lattices

Hennig

Karachalios

2023

Math Methods in App Sciences

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We prove the existence of periodic traveling wave solutions for general discrete nonlinear Klein‐Gordon systems, considering both cases of hard and soft on‐site potentials. In the case of hard on‐site potentials, we implement a fixed‐point theory approach, combining Schauder's fixed‐point theorem and the contraction mapping principle. This approach enables us to identify a ring in the energy space for nontrivial solutions to exist, energy (norm) thresholds for their existence, and upper bounds on their velocity. In the case of soft on‐site potentials, the proof of existence of periodic traveling wave solutions is facilitated by a variational approach based on the mountain pass theorem. Thresholds on the averaged kinetic energy for these solutions to exist are also derived.

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Traveling waves in Fermi-Pasta-Ulam lattices with saturable nonlinearities

Cited by 25 publications

References 15 publications

Wave Propagation in Diatomic Lattices

Wave Propagation in Diatomic Lattices

Periodic and Solitary Travelling Waves in Infinite Lattices without Ambrosetti–Rabinowitz Condition

Periodic traveling wave solutions of discrete nonlinear Klein‐Gordon lattices

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