Complex solitary waves and soliton trains in KdV and mKdV equations

Modak, Subhrajit; Singh, Ashutosh Kumar; Panigrahi, Prasanta K.

doi:10.1140/epjb/e2016-70130-7

Cited by 14 publications

(14 citation statements)

References 52 publications

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“…They can occur in diffusion-controlled unidirectional crystal growth [34]. One-and two-soliton complex solutions of the KdV equation have been found and related to PT -symmetric systems [35,36], quantum-mechanical scattering matrices in a semiclassical approximation [30], and to a number of other physical systems where the KdV is the governing equation.…”

Section: Discussionmentioning

confidence: 99%

Shallow-water rogue waves: An approach based on complex solutions of the Korteweg–de Vries equation

2019

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The formation of rogue waves in shallow water is presented in this Rapid Communication by providing the three lowest-order exact rational solutions to the Korteweg-de Vries (KdV) equation. They have been obtained from the modified KdV equation by using the complex Miura transformation. It is found that the amplitude amplification factor of such waves formed in shallow water is much larger than the amplitude amplification factor of those occurring in deep water. These solutions clearly demonstrate a potential hazard for coastal areas. They can also provide a solid mathematical basis for the existence of abnormally large-amplitude waves in other branches of nonlinear physics such as optics, unidirectional crystal growth, and in quantum mechanics.

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Section: Discussionmentioning

confidence: 99%

Shallow-water rogue waves: An approach based on complex solutions of the Korteweg–de Vries equation

2019

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“…For simplicity, we take x 0 = 0, which retains the symmetric nature of potential at t = 0. The KdV equation is known to possess breather 50,51 and complex soliton 33,34,48 solutions. Among the prior, the Akhmediev breathers 52-54 are periodic in space and localized in time whereas Ma breathers 55 have the opposite behavior.…”

Section: Complex Kdv Breathers and Solitonsmentioning

confidence: 99%

“…Given a potential solves the KdV system, the corresponding superpotential satisfies the modified KdV (mKdV) equations 33,45 as the respective solutions to these two equations are related through the Miura transformation: u = v 2 ± v x 46, 47 . Moreover, since there are two distinct mKdV equations with solutions connected as v → iv, there is another class of KdV solutions with functional form: u = −v 2 ± iv x 46, 48 .…”

Section: Introductionmentioning

confidence: 99%

PT-symmetric KdV Solutions and Their Algebraic Extension with Zero-Width Resonances

Abhinav,

Shukla,

Panigrahi

2024

Preprint

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A class of complex breather and soliton solutions to both KdV and mKdV equations are identified with a Pöschl-Teller type PT-symmetric potential. However, these solutions represent only the unbroken-PT phase owing to their isospectrality to an infinite potential well in the complex plane having real spectra. To obtain the broken-PT phase, an extension of the potential satisfying the sl (2,ℝ) potential algebra is mandatory that additionally supports non-trivial zero-width resonances.

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“…For the real Scarf II, φ(x) satisfies the modified KdV equation [83,84]. In the case of complex superpotential potential, W 1 (x) = W 2 (x) + φ(x), where…”

Section: (A) Isospectral Deformation and Connection With Modified Kdv Equationmentioning

confidence: 99%

PT-symmetry and supersymmetry: interconnection of broken and unbroken phases

et al. 2021

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The broken and unbroken phases of P T and supersymmetry in optical systems are explored for a complex refractive index profile in the form of a Scarf potential, under the framework of supersymmetric quantum mechanics. The transition from unbroken to the broken phases of P T -symmetry, with the merger of eigenfunctions near the exceptional point is found to arise from two distinct realizations of the potential, originating from the underlying supersymmetry. Interestingly, in P T -symmetric phase, spontaneous breaking of supersymmetry occurs in a parametric domain, possessing non-trivial shape invariances, under reparametrization to yield the corresponding energy spectra. One also observes a parametric bifurcation behaviour in this domain. Unlike the real Scraf potential, in P T -symmetric phase, a connection between complex isospecrtal superpotentials and modified Korteweg-de Vries equation occurs, only with certain restrictive parametric conditions. In the broken P T -symmetry phase, supersymmetry is found to be intact in the entire parameter domain yielding the complex energy spectra, with zero-width resonance occurring at integral values of a potential parameter.

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Complex solitary waves and soliton trains in KdV and mKdV equations

Cited by 14 publications

References 52 publications

Shallow-water rogue waves: An approach based on complex solutions of the Korteweg–de Vries equation

Shallow-water rogue waves: An approach based on complex solutions of the Korteweg–de Vries equation

PT-symmetric KdV Solutions and Their Algebraic Extension with Zero-Width Resonances

PT-symmetry and supersymmetry: interconnection of broken and unbroken phases

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