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

Gaillard, Pierre

doi:10.22606/nhmp.2017.11004

Cited by 2 publications

(3 citation statements)

References 14 publications

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“…In the (x, y) plane of coordinates, different structures appear. All the solutions described in this study are different from those constructed in previous works [26][27][28][29][30][31] .…”

Section: Discussionmentioning

confidence: 59%

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Multi-Parametric Families of Real and Non Singular Solutions of the Kadomtsev-Petviasvili I Equation

Gaillard

2021

Universal Journal of Mathematics and Applications

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Multi-parametric solutions to the Kadomtsev-Petviashvili equation (KPI) in terms of Fredholm determinants are constructed in function of exponentials. A representation of these solutions as a quotient of wronskians of order 2N in terms of trigonometric functions is deduced. All these solutions depend on 2N − 1 real parameters. A third representation in terms of a quotient of two real polynomials depending on 2N − 2 real parameters is given; the numerator is a polynomial of degree 2N(N + 1) − 2 in x, y and t and the denominator is a polynomial of degree 2N(N + 1) in x, y and t. The maximum absolute value is equal to 2(2N + 1) 2 − 2. We explicitly construct the expressions for the first third orders and we study the patterns of their absolute value in the plane (x, y) and their evolution according to time and parameters. It is relevant to emphasize that all these families of solutions are real and non singular.

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Section: Discussionmentioning

confidence: 59%

“…This type of solution to the KPI equation is different from our previous works. In our previous works [26][27][28][29][30][31], we constructed solution of order 1 to KPI equation and got ṽ1 (X,Y,…”

Section: Case N =mentioning

confidence: 99%

Multi-Parametric Families of Real and Non Singular Solutions of the Kadomtsev-Petviasvili I Equation

Gaillard

2021

Universal Journal of Mathematics and Applications

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“…Here we have given a new method to construct solutions to the Johnson equation related to previous results [12][13][14].…”

Section: Resultsmentioning

confidence: 99%

The Johnson Equation, Fredholm and Wronskian Representations of Solutions, and the Case of Order Three

Gaillard

2018

Advances in Mathematical Physics

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We construct solutions to the Johnson equation (J) first by means of Fredholm determinants and then by means of Wronskians of order 2N giving solutions of order N depending on 2N-1 parameters. We obtain N order rational solutions that can be written as a quotient of two polynomials of degree 2N(N+1) in x, t and 4N(N+1) in y depending on 2N-2 parameters. This method gives an infinite hierarchy of solutions to the Johnson equation. In particular, rational solutions are obtained. The solutions of order 3 with 4 parameters are constructed and studied in detail by means of their modulus in the (x,y) plane in function of time t and parameters a1, a2, b1, and b2.

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

Cited by 2 publications

References 14 publications

Multi-Parametric Families of Real and Non Singular Solutions of the Kadomtsev-Petviasvili I Equation

Multi-Parametric Families of Real and Non Singular Solutions of the Kadomtsev-Petviasvili I Equation

The Johnson Equation, Fredholm and Wronskian Representations of Solutions, and the Case of Order Three

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