KdV breathers on a cnoidal wave background

Hoefer, Mark; Mucalica, Ana; Pelinovsky, Dmitry E.

doi:10.1088/1751-8121/acc6a8

Cited by 14 publications

(5 citation statements)

References 26 publications

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“…In fact, soliton–cnoidal wave interaction in the KdV equation was studied some time ago 30,90 in which exact solutions corresponding to N soliton “dislocations” to a cnoidal wave were obtained. For the

N=1

case, we recognize these solutions as breathers, bright or dark, exhibiting two time scales associated with their propagation and background oscillations 30,31 . As we will see, breathers play an important role in soliton–DSW interaction.…”

Section: Introductionmentioning

confidence: 75%

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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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N=1

Section: Introductionmentioning

confidence: 75%

“…For the 𝑁 = 1 case, we recognize these solutions as breathers, bright or dark, exhibiting two time scales associated with their propagation and background oscillations. 30,31 As we will see, breathers play an important role in soliton-DSW interaction.…”

Section: Introductionmentioning

confidence: 93%

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

Ablowitz

Cole

et al. 2023

Stud Appl Math

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“…Finding conditions when this continuity is satisfied is an interesting problem for future studies. We think that our method can be generalized straightforwardly to multibreather solutions, breathers on a nontrivial background (e.g., cnoidal waves), and other integrable systems including vector breathers, [40][41][42][43][44][45][46] making it possible to model the rich dynamics of complex breather interactions 24,62,63,[69][70][71][72] and the behavior of breathers in nearly integrable systems. 73,74 We assume that for such a generalization one will only need to find the solitonic model for the breather background.…”

Section: Conclusion and Discussionmentioning

confidence: 99%

“…We believe that our method can be applied straightforwardly to general multibreather solutions, breathers on a nontrivial background (e.g., cnoidal waves), and other integrable systems including vector breathers. [40][41][42][43][44][45][46] The paper is organized as follows. In Section 2, we discuss the DM procedure, the multisoliton and multibreather solutions, and the solitonic model of the plane wave.…”

Section: Introductionmentioning

confidence: 99%

“…At large N , the constructed solutions are practically indistinguishable from breathers in a wide region of space and time. We believe that our method can be applied straightforwardly to general multibreather solutions, breathers on a nontrivial background (e.g., cnoidal waves), and other integrable systems including vector breathers 40–46 …”

Section: Introductionmentioning

confidence: 99%

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Multisoliton interactions approximating the dynamics of breather solutions

Agafontsev,

Gelash,

Randoux

et al. 2023

Stud Appl Math

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Nowadays, breather solutions are generally accepted models of rogue waves. However, breathers exist on a finite background and therefore are not localized, while wavefields in nature can generally be considered as localized due to the limited sizes of physical domain. Hence, the theory of rogue waves needs to be supplemented with localized solutions, which evolve locally as breathers. In this paper, we present a universal method for constructing such solutions from exact multisoliton solutions, which consists in replacing the plane wave in the dressing construction of the breathers with a specific exact N‐soliton solution converging asymptotically to the plane wave at large number of solitons N. On the example of the Peregrine, Akhmediev, Kuznetsov–Ma, and Tajiri–Watanabe breathers, we show that constructed with our method multisoliton solutions, being localized in space with characteristic width proportional to N, are practically indistinguishable from the breathers in a wide region of space and time at large N. Our method makes it possible to build solitonic models with the same dynamical properties for the higher order rational and super‐regular breathers, and can be applied to general multibreather solutions, breathers on a nontrivial background (e.g., cnoidal waves), and other integrable systems. The constructed multisoliton solutions can also be generalized to capture the spontaneous emergence of rogue waves through the spontaneous synchronization of soliton norming constants, though finding these synchronization conditions represents a challenging problem for future studies.

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KP solitons and the Riemann theta functions

Kodama

2024

Lett Math Phys

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We show that the $$\tau $$ τ -functions of the regular KP solitons from the totally nonnegative Grassmannians can be expressed by the Riemann theta functions on singular curves. We explicitly write the parameters in the Riemann theta function in terms of those of the KP soliton. We give a short remark on the Prym theta function on a double covering of singular curves. We also discuss the KP soliton on quasi-periodic background, which is obtained by applying the vertex operators to the Riemann theta function.

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KdV breathers on a cnoidal wave background

Cited by 14 publications

References 26 publications

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

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

Multisoliton interactions approximating the dynamics of breather solutions

KP solitons and the Riemann theta functions

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