Strongly interacting multi-solitons with logarithmic relative distance for the gKdV equation

Vinh, Nguyễn Tiến

doi:10.1088/1361-6544/aa8cab

Cited by 14 publications

(4 citation statements)

References 30 publications

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“…Solitons associated to higher-order poles have also been studied for the modified Korteweg-de Vries equation [35], the sine-Gordon equation [3,25,32], the Caudrey-Dodd-Gibbon-Sawada-Kotera equation [15], the Kadomtsev-Petviashvili I equation [1,33], the N -wave system [28], the complex short-pulse equation [19], and the coupled Sasa-Satsuma system [18]. Vinh [34] has recently studied analogues of higher-order solitons for nonintegrable generalized Korteweg-de Vries equations.…”

Section: Introductionmentioning

confidence: 99%

Large-Order Asymptotics for Multiple-Pole Solitons of the Focusing Nonlinear Schrödinger Equation

2019

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The integrable focusing nonlinear Schrödinger equation admits soliton solutions whose associated spectral data consist of a single pair of conjugate poles of arbitrary order. We study families of such multiple-pole solitons generated by Darboux transformations as the pole order tends to infinity. We show that in an appropriate scaling, there are four regions in the spacetime plane where solutions display qualitatively distinct behaviors: an exponential-decay region, an algebraic-decay region, a non-oscillatory region, and an oscillatory region. Using the nonlinear steepest-descent method for analyzing Riemann-Hilbert problems, we compute the leading-order asymptotic behavior in the algebraic-decay, non-oscillatory, and oscillatory regions.

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Section: Introductionmentioning

confidence: 99%

Large-Order Asymptotics for Multiple-Pole Solitons of the Focusing Nonlinear Schrödinger Equation

2019

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“…As [1], there are several results about multisolitary waves with logarithmic distance; see e.g. [18,36,37,38]. We note that for the damped nonlinear Klein-Gordon equation, multi-solitary waves of (DNKG) are impossible to move at a constant speed by the damping α.…”

Section: Previous Resultsmentioning

confidence: 82%

Global Dynamics Around 2-Solitons for the Nonlinear Damped Klein-Gordon Equations

Kenjiro

Nakanishi

2022

Ann. PDE

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We consider the damped nonlinear Klein-Gordon equation with a delta potential, where p > 2, α > 0, γ < 2, and δ 0 = δ 0 (x) denotes the Dirac delta with the mass at the origin. When γ = 0, Côte, Martel and Yuan [7] proved that any global solution either converges to 0 or to the sum of K ≥ 1 decoupled solitary waves which have alternative signs. In this paper, we first prove that any global solution either converges to 0 or to the sum of K ≥ 1 decoupled solitary waves. Next we construct a single solitary wave solution that moves away from the origin when γ < 0 and construct an even 2-solitary wave solution when γ ≤ −2. Last we give single solitary wave solutions and even 2-solitary wave solutions an upper bound for the distance between the origin and the solitary wave.

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“…Other related results. The first construction of a two-soliton solution with trajectories having asymptotically vanishing velocities was obtained by Krieger, Martel and Raphaël [25], see also [33,43] for other constructions and [52] for related computations in the completely integrable setting. Existence of strongly interacting multi-solitons with an arbitrary number of solitons was obtained by Lan and Wang [27] for the generalized Benjamin-Ono equation.…”

Section: 4mentioning

confidence: 99%

Dynamics of kink clusters for scalar fields in dimension 1+1

Jendrej¹,

Lawrie²

2023

Preprint

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We consider a real scalar field equation in dimension 1 `1 with an even positive self-interaction potential having two non-degenerate zeros (vacua) 1 and ´1. It is known that such a model admits non-trivial static solutions called kinks and antikinks. A kink cluster is a solution approaching, for large positive times, a superposition of alternating kinks and antikinks whose velocities converge to 0. They can be equivalently characterised as the solutions of minimal possible energy containing a given number of transitions between the vacua, or as the solutions whose kinetic energy decays to 0 for large time.Our main result is a determination of the main-order asymptotic behaviour of any kink cluster. Moreover, we construct a kink cluster for any prescribed initial positions of the kinks and antikinks, provided that their mutual distances are sufficiently large. Finally, we show that kink clusters are universal profiles for the formation/collapse of multi-kink configurations. The proofs rely on a reduction, using appropriately chosen modulation parameters, to an n-body problem with attractive exponential interactions.

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Strongly interacting multi-solitons with logarithmic relative distance for the gKdV equation

Cited by 14 publications

References 30 publications

Large-Order Asymptotics for Multiple-Pole Solitons of the Focusing Nonlinear Schrödinger Equation

Large-Order Asymptotics for Multiple-Pole Solitons of the Focusing Nonlinear Schrödinger Equation

Global Dynamics Around 2-Solitons for the Nonlinear Damped Klein-Gordon Equations

Dynamics of kink clusters for scalar fields in dimension 1+1

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