Decay of Kadomtsev–Petviashvili lumps in dissipative media

Clarke, Simon R; Gorshkov, K. A.; Grimshaw, Roger; Stepanyants, Yury

doi:10.1016/j.physd.2017.11.009

Cited by 13 publications

(7 citation statements)

References 24 publications

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“…A small dissipation can cause a gradual decay of soliton histograms and their distortion, in general. The histogram decay and its distortion depend on the specific type of dissipation; this problem has not been studied yet, although the decay of individual solitons under the influence of various types of dissipation has been investigated for the KdV [12] and Benjamin-Ono [13] solitons, as well as for the Kadomtsev-Petviashvili lumps [4].…”

Section: Discussionmentioning

confidence: 99%

Soliton spectra of random water waves in shallow basins

Giovanangeli

Kharif

Stepanyants

2018

Math. Model. Nat. Phenom.

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Interpretation of random wave field on a shallow water in terms of Fourier spectra is not adequate, when wave amplitudes are not infinitesimally small. A nonlinearity of wave fields leads to the harmonic interactions and random variation of Fourier spectra. As has been shown by Osborne and his co-authors, a more adequate analysis can be performed in terms of nonlinear modes representing cnoidal waves; a spectrum of such modes remains unchanged even in the process of nonlinear mode interactions. Here we show that there is an alternative and more simple analysis of random wave fields on shallow water, which can be presented in terms of interacting Korteweg-de Vries solitons. The data processing of random wave field is developed on the basis of inverse scattering method. The soliton component obscured in a random wave field is determined and a corresponding distribution function of number of solitons on their amplitudes is constructed.The approach developed is illustrated by means of artificially generated quasi-random wave field and applied to the real data interpretation of wind waves generated in the laboratory wind tank.

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Section: Discussionmentioning

confidence: 99%

Soliton spectra of random water waves in shallow basins

Giovanangeli

Kharif

Stepanyants

2018

Math. Model. Nat. Phenom.

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“…Meanwhile, the approximate asymptotic method earlier developed for the description of mainly one-dimensional soliton interactions (see, for example, [5][6][7]) can be extended to the two-dimensional case too [8]. In particular, such an approach has been used for the description of two-dimensional lumps [9,10].…”

Section: Exact Two-soliton Solution Of the Kadomtsev-petviashvili Equmentioning

confidence: 99%

“…For the description of soliton interactions, various asymptotic approaches have been developed basically for the one-dimensional cases (see, for example, [5][6][7] and references therein). There are a few developments of asymptotic methods for the two-dimensional space for the description of plane soliton interactions [8] and for the description of fully localized two-dimensional lumps [9,10]. Here we suggest the asymptotic method for the description of symmetric soliton patterns applicable for a wide class of quasi-one-dimensional equations of the KP-type.…”

Section: The Approximate Asymptotic Approach To the Description Of Twmentioning

confidence: 99%

The Asymptotic Approach to the Description of Two-Dimensional Symmetric Soliton Patterns

Stepanyants

2020

Symmetry

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The asymptotic approach is suggested for the description of interacting surface and internal oceanic solitary waves. This approach allows one to describe stationary moving symmetric wave patterns consisting of two plane solitary waves of equal amplitudes moving at an angle to each other. The results obtained within the approximate asymptotic theory are validated by comparison with the exact two-soliton solution of the Kadomtsev–Petviashvili equation (KP2-equation). The suggested approach is equally applicable to a wide class of non-integrable equations too. As an example, the asymptotic theory is applied to the description of wave patterns in the 2D Benjamin–Ono equation describing internal waves in the infinitely deep ocean containing a relatively thin one of the layers.

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“…Obviously, for sufficiently large (small) surface tension, we can attain KP‐I (KP‐II) equation, respectively. In addition, the KP equation is also applicable in several fields, such as plasma physics, solid state physics, and thin plates physics 5–10 …”

Section: Introductionmentioning

confidence: 99%

“…In addition, the KP equation is also applicable in several fields, such as plasma physics, solid state physics, and thin plates physics. [5][6][7][8][9][10] Both KP-I and KP-II equations possess line solitons. [11][12][13][14] Moreover, the multiline solitons of the KP-II equation can also display resonant collisions.…”

Section: Introductionmentioning

confidence: 99%

Completely resonant collision of lumps and line solitons in the Kadomtsev–Petviashvili I equation

et al. 2021

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Resonant collisions among localized lumps and line solitons of the Kadomtsev–Petviashvili I (KP‐I) equation are studied. The KP‐I equation describes the evolution of weakly nonlinear, weakly dispersive waves with slow transverse variations. Lumps can only exist for the KP equation when the signs of the transverse derivative and the weak dispersion in the propagation direction are different, that is, in the KP‐I regime. Collisions among lumps and solitons for “integrable” equations are normally elastic, that is, wave shapes are preserved except possibly for phase shifts. For resonant collisions, mathematically the phase shift will become indefinitely large. Physically a lump may be detached (or emitted) from a line soliton, survives for a brief transient period in time, and then merges with the next adjacent soliton. This special lump is thus localized in time as well as the two spatial dimensions, and can be termed a “rogue lump.” By employing a reduction method for the KP hierarchy in conjunction with the Hirota bilinear technique, a general L‐lump/(L+1)‐line‐soliton solution is obtained for the KP‐I equation for an arbitrary positive integer L. For L≥2, collisions among modes become increasingly complicated, with multiple lumps detaching from one single or different line solitons and then disappearing into the remaining solitons. In terms of applications, such “rogue lumps” represent a mode truly localized both in space and time, and will be valuable in modeling physical problems.

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Decay of Kadomtsev–Petviashvili lumps in dissipative media

Cited by 13 publications

References 24 publications

Soliton spectra of random water waves in shallow basins

Soliton spectra of random water waves in shallow basins

The Asymptotic Approach to the Description of Two-Dimensional Symmetric Soliton Patterns

Completely resonant collision of lumps and line solitons in the Kadomtsev–Petviashvili I equation

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