Solitons on the rarefaction wave background via the Darboux transformation

Mucalica, Ana; Pelinovsky, Dmitry E.

doi:10.1098/rspa.2022.0474

Cited by 5 publications

(7 citation statements)

References 41 publications

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“…has recently been shown that the spectrum for the trapped, pseudosoliton for the soliton-RW interaction is identified by a resonant pole in the complex plane. 27 There are a number of implications of Equation ( 82), which we now investigate. First, we need to distinguish between where the soliton center 𝑥 0 is initialized.…”

Section: Soliton-rw Interactionmentioning

confidence: 99%

“…In Section 4, we confirm by IST analysis that λ 0 defined in () for soliton–RW and soliton–DSW interaction from modulation theory is precisely the proper eigenvalue for a genuine soliton solution or a pseudoembedded eigenvalue when a genuine soliton solution does not exist. It has recently been shown that the spectrum for the trapped, pseudosoliton for the soliton–RW interaction is identified by a resonant pole in the complex plane 27 …”

Section: Whitham Modulation Theorymentioning

confidence: 99%

“…Another IST‐related approach that leverages the integrability of the KdV equation is the Darboux transformation, which was used in Ref. [27] to obtain a nonlinear superposition of a soliton and RW at

t = 0

for the transmission case.…”

Section: Introductionmentioning

confidence: 99%

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Soliton–mean field interaction in Korteweg–de Vries dispersive hydrodynamics

Ablowitz

Cole

et al. 2023

Stud Appl Math

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The mathematical description of localized solitons in the presence of large‐scale waves is a fundamental problem in nonlinear science, with applications in fluid dynamics, nonlinear optics, and condensed matter physics. Here, the evolution of a soliton as it interacts with a rarefaction wave or a dispersive shock wave, examples of slowly varying and rapidly oscillating dispersive mean fields, for the Korteweg–de Vries equation is studied. Step boundary conditions give rise to either a rarefaction wave (step up) or a dispersive shock wave (step down). When a soliton interacts with one of these mean fields, it can either transmit through (tunnel) or become embedded (trapped) inside, depending on its initial amplitude and position. A topical review of three separate analytical approaches is undertaken to describe these interactions. First, a basic soliton perturbation theory is introduced that is found to capture the solution dynamics for soliton–rarefaction wave interaction in the small dispersion limit. Next, multiphase Whitham modulation theory and its finite‐gap description are used to describe soliton–rarefaction wave and soliton–dispersive shock wave interactions. Lastly, a spectral description and an exact solution of the initial value problem is obtained through the inverse scattering transform. For transmitted solitons, far‐field asymptotics reveal the soliton phase shift through either type of wave mentioned above. In the trapped case, there is no proper eigenvalue in the spectral description, implying that the evolution does not involve a proper soliton solution. These approaches are consistent, agree with direct numerical simulation, and accurately describe different aspects of solitary wave–mean field interaction.

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Section: Soliton-rw Interactionmentioning

confidence: 99%

Section: Whitham Modulation Theorymentioning

confidence: 99%

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Soliton–mean field interaction in Korteweg–de Vries dispersive hydrodynamics

Ablowitz

Cole

et al. 2023

Stud Appl Math

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“…When a soliton becomes embedded or trapped within a DSW, the trapped soliton resembles a depression (dark) nonlinear wavepacket. Similar transmission and trapping scenarios were analyzed for solitons interacting with rarefaction waves [7,8].…”

Section: Introductionmentioning

confidence: 99%

KdV breathers on a cnoidal wave background

Hoefer¹,

Mucalica²,

Pelinovsky³

2023

Preprint

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Using the Darboux transformation for the Korteweg-de Vries equation, we construct and analyze exact solutions describing the interaction of a solitary wave and a traveling cnoidal wave. Due to their unsteady, wavepacket-like character, these wave patterns are referred to as breathers. Both elevation (bright) and depression (dark) breather solutions are obtained. The nonlinear dispersion relations demonstrate that the bright (dark) breathers propagate faster (slower) than the background cnoidal wave. Two-soliton solutions are obtained in the limit of degeneration of the cnoidal wave. In the small amplitude regime, the dark breathers are accurately approximated by dark soliton solutions of the nonlinear Schrödinger equation. These results provide insight into recent experiments on soliton-dispersive shock wave interactions and soliton gases.

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“…Recently, the interaction between an individual (tracer) soliton and a large-scale coherent nonlinear structure such as rarefaction and dispersive shock wave has been studied both theoretically and experimentally [22][23][24][25][26][27][28]. The trajectory of the tracer soliton within these macroscopic nonlinear structures is directed by the structure's mean field, which results either in the trapping of the soliton inside or its transmission (tunneling) through the large-scale nonlinear wave.…”

mentioning

confidence: 99%

Soliton refraction through an optical soliton gas

Suret¹,

Dufour²,

Roberti³

et al. 2023

Preprint

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We report an optical fiber experiment in which we investigate the interaction between an individual soliton and a dense soliton gas. We evidence a refraction phenomenon where the tracer soliton experiences an effective velocity change due to its interaction with the optical soliton gas. This interaction results in a significant spatial shift that is measured and compared with theoretical predictions obtained via the inverse scattering transform (IST) theory. The effective velocity change associated with the refraction phenomenon is found to be in good quantitative agreement with the results of the spectral kinetic theory of soliton gas. Our results validate the collision rate ansatz that plays a fundamental role in the kinetic theory of soliton gas and also is at heart of generalized hydrodynamics of many-body integrable systems.

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Solitons on the rarefaction wave background via the Darboux transformation

Cited by 5 publications

References 41 publications

Soliton–mean field interaction in Korteweg–de Vries dispersive hydrodynamics

Soliton–mean field interaction in Korteweg–de Vries dispersive hydrodynamics

KdV breathers on a cnoidal wave background

Soliton refraction through an optical soliton gas

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