Two-dimensional solitons of the Kadomtsev-Petviashvili equation and their interaction

A recently observed relation between 'weakly nonassociative' algebras A (for which the associator (A, A 2 , A) vanishes) and the KP hierarchy (with dependent variable in the middle nucleus A ′ of A) is recalled. For any such algebra there is a nonassociative hierarchy of ODEs, the solutions of which determine solutions of the KP hierarchy. In a special case, and with A ′ a matrix algebra, this becomes a matrix Riccati hierarchy which is easily solved. The matrix solution then leads to solutions of the scalar KP hierarchy. We discuss some classes of solutions obtained in this way.

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“…. , c N ), u becomes an N -soliton solution of the scalar KP hierarchy [6][7][8]. For example, with M = N = 2,…”

Section: Some Solutions Of the Scalar Kp Hierarchymentioning

confidence: 99%

From nonassociativity to solutions of the KP hierarchy

Dimakis

Müller‐Hoissen

2006

Czech J Phys

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“…We use the following notations : (3) and parameters e ν defined by (4). We define the following functions :…”

Section: Rational Solutionsmentioning

confidence: 99%

“…Zakharov extended the inverse scattering transform (IST) to this KPI equation, and obtained several exact solutions. The first rational solutions were found in 1977 by Manakov, Zakharov, Bordag and Matveev [4]. Other researches were led and more general rational solutions to the KPI equation were obtained.…”

Section: Introductionmentioning

confidence: 99%

“…This method gives an infinite hierarchy of rational solutions of order N depending on 2N − 2 real parameters. We construct here the explicit rational solutions of order 5, depending on 8 real parameters, and the representations of their modulus in the plane of the coordinates (x, y) according to the real parameters a 1 4 , b 4 and time t.…”

Section: Introductionmentioning

confidence: 99%

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Families of Rational Solutions of Order 5 to the KPI Equation Depending on 8 Parameters

Gaillard¹

2017

NHMP

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In this paper, we go on with the study of rational solutions to the Kadomtsev-Petviashvili equation (KPI). We construct here rational solutions of order 5 as a quotient of 2 polynomials of degree 60 in x, y and t depending on 8 parameters. The maximum modulus of these solutions at order 5 is checked as equal to 2(2N + 1) 2 = 242. We study their modulus patterns in the plane (x, y) and their evolution according to time and parameters a1, a2, a3, a4, b1, b2, b3, b4. We get triangle and ring structures as obtained in the case of the NLS equation.

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“…The interested reader can consult [13,14]. The multi-soliton solution of the Kadomtsev-Petviashvili I (KPI) equation, that is needed in order to build the multi-soliton solution of the DSI and DSIII systems, has been derived in [15] and in its most general form in [16].…”

Section: Guidelines For Additional Readingmentioning

confidence: 99%

Multidimensional localized solitons

Boiti

Martina

Pempinelli

1995

Chaos, Solitons & Fractals

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Recently it has been discovered that some nonlinear evolution equations in 2 + 1 dimensions, which are integrable by the use of the Spectral Transform, admit localized (in the space) soliton solutions. This article briefly reviews some of the main results obtained in the last five years thanks to the renewed interest in soliton theory due to this discovery. The theoretical tools needed to understand the unexpected richness of behaviour of multidimensional localized solitons during their mutual scattering are furnished. Analogies and especially discrepancies with the unidimensional case are stressed.

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Two-dimensional solitons of the Kadomtsev-Petviashvili equation and their interaction

Cited by 557 publications

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From nonassociativity to solutions of the KP hierarchy

From nonassociativity to solutions of the KP hierarchy

Families of Rational Solutions of Order 5 to the KPI Equation Depending on 8 Parameters

Multidimensional localized solitons

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