Solitary Wave Solutions to a Class of Modified Green–Naghdi Systems

Duchêne, Vincent; Nilsson, Dag; Wahlén, Erik

doi:10.1007/s00021-017-0355-0

Cited by 15 publications

(23 citation statements)

References 40 publications

(76 reference statements)

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“…We proceed by performing a rigorous local variational reduction which converts I ε to a perturbation T ε of T 0 (Section 3). The estimate (9) suggests that the spectrum of a solitary wave u(x, y) is concentrated in the region |k 1 |, | k2 k1 | 1. We therefore decompose u into the sum of functions u 1 and u 2 whose spectra are supported in the region…”

Section: Introductionmentioning

confidence: 95%

Small-amplitude fully localised solitary waves for the full-dispersion Kadomtsev–Petviashvili equation

Ehrnström

Groves

2018

Nonlinearity

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The KP-I equation (ut − 2uux + 1 2 (β − 1 3 )uxxx)x − uyy = 0 arises as a weakly nonlinear model equation for gravity-capillary waves with strong surface tension (Bond number β > 1/3). This equation admits -as an explicit solution -a 'fully localised' or 'lump' solitary wave which decays to zero in all spatial directions. Recently there has been interest in the full-dispersion KP-I equation ut + m(D)ux + 2uux = 0, where m(D) is the Fourier multiplier with symbol m(k) = 1 + β|k| 2 | 1 2 tanh |k| |k| 1 2 1 + 2k 2 2 k 2 1 1 2, which is obtained by retaining the exact dispersion relation from the waterwave problem. In this paper we show that the FDKP-I equation also has a fully localised solitary-wave solution. The existence theory is variational and perturbative in nature. A variational principle for fully localised solitary waves is reduced to a locally equivalent variational principle featuring a perturbation of the variational functional associated with fully localised solitary-wave solutions of the KP-I equation. A nontrivial critical point of the reduced functional is found by minimising it over its natural constraint set.

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Section: Introductionmentioning

confidence: 95%

Small-amplitude fully localised solitary waves for the full-dispersion Kadomtsev–Petviashvili equation

Ehrnström

Groves

2018

Nonlinearity

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“…Such an explicit formula is of course unexpected for the fully dispersive system (WGN). However, the following result has been shown in [27]:…”

Section: Introductionmentioning

confidence: 99%

“…We choose the SGN and WGN equations as our subject of investigations in order to step out of the world of unidirectional scalar (nonlinear and dispersive) equations for which similar studies have been realized [1,41,42,52,50], while retaining strong structural properties. In particular, solitary waves can be identified with critical points of functionals which directly derive from the aforementioned Hamiltonian structure [27]. Moreover, the two systems of equations can (and will) be numerically approximated using identical numerical strategies, specifically Fourier pseudospectral methods.…”

Section: Introductionmentioning

confidence: 99%

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Numerical study of the Serre-Green-Naghdi equations and a fully dispersive counterpart

Duchêne

Klein

2022

DCDS-B

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<p style='text-indent:20px;'>We perform numerical experiments on the Serre-Green-Naghdi (SGN) equations and a fully dispersive "Whitham-Green-Naghdi" (WGN) counterpart in dimension 1. In particular, solitary wave solutions of the WGN equations are constructed and their stability, along with the explicit ones of the SGN equations, is studied. Additionally, the emergence of modulated oscillations and the possibility of a blow-up of solutions in various situations is investigated.</p><p style='text-indent:20px;'>We argue that a simple numerical scheme based on a Fourier spectral method combined with the Krylov subspace iterative technique GMRES to address the elliptic problem and a fourth order explicit Runge-Kutta scheme in time allows to address efficiently even computationally challenging problems.</p>

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“…Moreover, in [6,7,16], other types of fully dispersive Whitham-Boussinesq systems are considered. We also mention the generalized class of Green-Nagdhi equations introduced in [8], which was shown to posses solitary wave solutions in [9].…”

Section: Introductionmentioning

confidence: 99%

Solitary wave solutions to a class of Whitham–Boussinesq systems

Nilsson

Wang

2019

Z. Angew. Math. Phys.

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In this note we study solitary wave solutions of a class of Whitham-Boussinesq systems which includes the bi-directional Whitham system as a special example. The travelling wave version of the evolution system can be reduced to a single evolution equation, similar to a class of equations studied by Ehrnström, Groves and Wahlén [10]. In that paper the authors prove the existence of solitary wave solutions using a constrained minimization argument adapted to noncoercive functionals, developed by Buffoni [3], Groves and Wahlén [15], together with the concentration-compactness principle.2010 Mathematics Subject Classification. 76B15; 76B25, 35S30, 35A20.

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Solitary Wave Solutions to a Class of Modified Green–Naghdi Systems

Cited by 15 publications

References 40 publications

Small-amplitude fully localised solitary waves for the full-dispersion Kadomtsev–Petviashvili equation

Small-amplitude fully localised solitary waves for the full-dispersion Kadomtsev–Petviashvili equation

Numerical study of the Serre-Green-Naghdi equations and a fully dispersive counterpart

Solitary wave solutions to a class of Whitham–Boussinesq systems

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