On a wave field transformation described by the two-dimensional Kadomtsev-Petviashvili equation

Gorshkov, K. A.; Pelinovsky, D E

doi:10.1088/0266-5611/11/3/008

Cited by 9 publications

(23 citation statements)

References 11 publications

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“…Otherwise, the asymptotic behavior of the peaks remain essentially the same as above. (b) The two-lump dynamics was discussed in earlier studies [16,39] as well. But a more detailed discussion of the asymptotic dynamics of the two-lump solution including the time evolution of the peak heights, are presented in this paper.…”

Section: Two-lump Solutionmentioning

confidence: 94%

“…Furthermore, the peak amplitudes evolve in time and reaches a constant asymptotic value which equals that of the simple one-lump peak. It has been also known [16,22,36] that the dynamics of the KPI lumps are related to the multi-particle Calogero-Moser system; this connection was further explored in [30,31].…”

Section: Introductionmentioning

confidence: 99%

“…Yet another class of KPI rational solutions arise as bound states corresponding to eigenvalues with multiplicities associated with the time-dependent Schrödinger equation, and are also amenable to IST methodology [3,39]. These are called the multi-lump solutions which were originally found in [19] by algebraic techniques and further investigated by several authors [4,16,29]. The multi-lump solution is an ensemble of a finite number of localized structures (or peaks) interacting in a non-trivial manner unlike the n-simple lump solution.…”

Section: Introductionmentioning

confidence: 99%

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Dynamics of KPI lumps

Chakravarty¹,

Zowada²

2022

J. Phys. A: Math. Theor.

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A family of nonsingular rational solutions of the Kadomtsev-Petviashvili (KP) I equation are investigated. These solutions have multiple peaks whose heights are time-dependent and the peak trajectories in the xy-plane are altered after collision. Thus they differ from the standard multi-peaked KPI simple n-lump solutions whose peak heights as well as peak trajectories remain unchanged after interaction.The anomalous scattering occurs due to a non-trivial internal dynamics among the peaks in a slow time scale. This phenomena is explained by relating the peak locations to the roots of complex heat polynomials. It follows from the long time asymptotics of the solutions that the peak trajectories separate as O(√|t|) as |t| → ∞, and all the peak heights approach the same constant value corresponding to that of the simple 1-lump solution. Consequently, a multi-peaked n-lump solution evolves to a superposition of n 1-lump solutions asymptotically as |t| →∞.

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Section: Two-lump Solutionmentioning

confidence: 94%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Dynamics of KPI lumps

Chakravarty¹,

Zowada²

2022

J. Phys. A: Math. Theor.

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“…Recently, there have been several studies investigating high-order KP-I lumps employing different methods [7][8][9][10][11]. Groshkov studied the non-trivial interaction and anomalous scattering of two fundamental lumps from the second-order lump solution by using exact and approximate methods [3]. Ablowitz derived the high-order lump solutions using the inverse scattering method and the binary Darboux transformation [7].…”

Section: Introductionmentioning

confidence: 99%

“…These solutions show a trivial interaction that fundamental lumps consists of n fundamental lumps with different asymptotic velocities. Their trajectories remain unchanged before and after the collision process at large time, which can be regarded as the case of n simple poles [3]. Additionally, Arkadiev derived the multi-lump interaction on zero background from the DS-II equation using the inverse scattering transformation [4].…”

Section: Introductionmentioning

confidence: 99%

Prediction of general high-order lump solutions in the Davey–Stewartson II equation

Yan,

Long,

Chen

2023

Proc. R. Soc. A.

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Under investigation in this work is the Davey–Stewartson (DS) II equation. Based on the Kadomtsev–Petviashvili (KP) reduction method and Schur polynomial theory, we construct the general high-order lump solutions. The prediction solutions consisting of fundamental lumps and their positions are derived by extracting leading-order asymptotics of the Schur polynomials of true solutions. When indexes of the solutions are chosen as different positive integer combinations, the prediction solutions at large time reflect two classes of lump patterns of the true solutions. The first class of lump pattern with triangle shape is analytically described by root structure of the Yablonskii–Vorob’ev polynomial. When time t evolves from large negative to large positive, the triangle lump reverses itself along the y -direction. The second class of lump pattern consists of non-triangle in outer region, which is analytically described by non-zero root structure of the Wronskian–Hermit polynomial, together with possible triangle in the inner region, which is analytically described by root structure of the Yablonskii–Vorob’ev polynomial. In addition, the non-triangle lump pattern in outer regions rotates at an angle while the possible triangle lump pattern in the inner region reverses itself along the y -direction when time t evolves from large negative to large positive. The obtained results improve our understanding of time evolution mechanisms of high-order lumps.

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An asymptotic approach to solitary wave instability and critical collapse in long-wave KdV-type evolution equations

Pelinovsky

Grimshaw

1996

Physica D: Nonlinear Phenomena

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On a wave field transformation described by the two-dimensional Kadomtsev-Petviashvili equation

Cited by 9 publications

References 11 publications

Dynamics of KPI lumps

Dynamics of KPI lumps

Prediction of general high-order lump solutions in the Davey–Stewartson II equation

An asymptotic approach to solitary wave instability and critical collapse in long-wave KdV-type evolution equations

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