Painlevé analysis and reducibility to the canonical form for the generalized Kadomtsev–Petviashvili equation

Brugarino, T.; Greco, Antonio

doi:10.1063/1.529095

Cited by 16 publications

(7 citation statements)

References 13 publications

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“…The variablecoefficient generalizations of the KP equation (GvcKPs), however, are able to provide us with more realistic models in such dynamical situations as the canonical and cylindrical cases, propagation of surface waves in large channels of varying width and depth with nonvanishing vorticity, etc. See, e.g., [7].…”

Section: Yt Gao and B Tianmentioning

confidence: 99%

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On an analytical method and soliton-typed solutions for certain variable-coefficient partial differential equations in nonlinear mechanics

Gao

Tian

1998

Acta Mechanica

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Summary.Among the currently interesting nonlinear topics are the variable-coefficient generalizations of the Kadomtsev-Petviashvili equation (GvcKPs), which are of practical value in mechanical and physical sciences. In this note, we propose a generalized variable-coefficient tanh method with computerized symbolic manipulation for a GvcKP to construct certain soliton-typed analytical solutions. In illustration, the nearly concentric Korteweg-de Vries (KdV) equation is presented as an example, and the cylindrical KP equation as a counter example. Similar success not shown here is on the variable-coefficient KdV equations.Historically, the milestone Korteweg-de Vries (KdV) equation,was introduced in 1895 in the context of fluid dynamics [1] to deal with the evolution of gravity-induced one-dimensional waves on shallow water of uniform depth, or the wave motion in shallow canals. Since 1960's, in the computation of Eq. (1), the soliton has been discovered/ named [2], and an exciting branch of nonlinear sciences has emerged. Nowadays, Eq. (1) is considered as the prototype of nonlinear evolution equations. However, a real ocean, for example, is inhomogeneous and the dynamics of nonlinear waves is considerably influenced by refraction, geometric divergence, etc. In fact, a real wave is not one-dimensional because its parameters vary along the front. The problem of evolution of transverse perturbations of a front is of theoretical and practical interest. Since 1970, the Kadomtsev-Petviashvili (KP) equation [3] has been introduced,which is the two-dimensional KdV equation for weakly nonplanar waves, describing slow variations in the y-direction of wave propagation with dispersive and nonlinear effects in the x-direction. In particular, the single soliton solution of Eq. (1) is unstable in media with a positive dispersion (with the "-" sign in the last term of Eq. (2)). Examples are long-wavelength gravity-capillary waves on the surface of a liquid and certain magnetosonic waves in plasmas.In the other case, it is stable in media with a negative dispersion ("+" sign). Examples are long-wavelength gravity (shallow-water) waves and ion-acoustic waves in unmagnetized plasmas. Equation (2) has many mechanical and physical applications, including, e.g., weakly two-dimensional long waves in shallow water [4], [5].

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Section: Yt Gao and B Tianmentioning

confidence: 99%

“…In this note, aiming at analytical solutions, we try to extend the generalized tanh method originally for certain constant-coefficient equations [9], [10] to cover the following type of the GvcKPs [7]:…”

Section: Yt Gao and B Tianmentioning

confidence: 99%

On an analytical method and soliton-typed solutions for certain variable-coefficient partial differential equations in nonlinear mechanics

Gao

Tian

1998

Acta Mechanica

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“…It only makes sense for PDE's and ODE's of a very special form, namely those for which the concepts of "Bureau symbol" and "resonance" are defined (see Section 4 below) and for which the resonance numbers are positive integers. Most papers on Painleve-testing of PDE's restrict attention to PDE's of the required special form, for example, the Kadomtsev-Petviashvili equation with its constant coefficients replaced by arbitrary functions of the coordinates [66,67] or generalizations of the nonlinear Schrodinger equation [34] (the latter paper also contains more general expansions). HlavatY [68,69,70] We hope that the present paper and paper II will illustrate this fact and put the freedom from logarithms test in a truer perspective.…”

Section: Definition a Singularity Ofy(x) Is Said To Be Movable If Itmentioning

confidence: 99%

Painlevé Classification of All Semi linear Partial Differential Equations of the Second Order. I. Hyperbolic Equations in Two Independent Variables

Cosgrove

1993

Stud Appl Math

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In this paper we give a complete classification of all Painlevé‐type hyperbolic partial differential equations (PDE's) over the complex domain of the form uxy = F(x, y, u, ux, uy where F is rational in u, ux, and uy, and locally analytic in x and y. We find exactly 22 equivalence classes of equations (under coordinate changes and Mobius transformations in u), which we denote HS‐I, HS‐II,…, HS‐XXII. A canonical representative of each class is presented and solved by transforming it either to a well‐known soliton equation (sine‐Gordon, Bullough‐Dodd‐Mikhailov) or to a linear equation by means of a Bäcklund correspondence or simpler change of variables. (The parabolic case, in which 10 more canonical equations are obtained, and semilinear PDE's in three or more variables are treated in the accompanying paper II.) The proof that the list is complete involves investigating four sets of necessary conditions in turn, each of which has essentially new features peculiar to the PDE context, as well as familiar features analogous to the corresponding conditions for ordinary differential equations (ODE's) as discussed in the classical literature by Painleve, Gambier, Ince, Bureau, and others. In a setting sufficiently general to embrace ODE's, hyperbolic and parabolic PDE's, and higher dimensional semilinear PDE's, we classify 76 types of O/PDE's, denoted DE‐1,…, according to their Bureau symbols and resonance data, which satisfy those necessary conditions for the Painlevé property common to each of these four Painlevé classification problems.

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“…Therefore, equations with variable coefficients may provide various models for real physical phenomena, for example, in the propagation of small-amplitude surface waves, which runs on straits or large channels of slowly varying depth and width. On one hand, there has been much interest in the study of generalizations with variable coefficients of nonlinear integrable equations [24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45].…”

Section: Introductionmentioning

confidence: 99%

The Painlevé Test and Reducibility to the Canonical Forms for Higher-Dimensional Soliton Equations with Variable-Coefficients

Kobayashi

Toda

2006

SIGMA

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Abstract. The general KdV equation (gKdV) derived by T. Chou is one of the famous (1 + 1) dimensional soliton equations with variable coefficients. It is well-known that the gKdV equation is integrable. In this paper a higher-dimensional gKdV equation, which is integrable in the sense of the Painlevé test, is presented. A transformation that links this equation to the canonical form of the Calogero-Bogoyavlenskii-Schiff equation is found. Furthermore, the form and similar transformation for the higher-dimensional modified gKdV equation are also obtained.

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Painlevé analysis and reducibility to the canonical form for the generalized Kadomtsev–Petviashvili equation

Cited by 16 publications

References 13 publications

On an analytical method and soliton-typed solutions for certain variable-coefficient partial differential equations in nonlinear mechanics

On an analytical method and soliton-typed solutions for certain variable-coefficient partial differential equations in nonlinear mechanics

Painlevé Classification of All Semi linear Partial Differential Equations of the Second Order. I. Hyperbolic Equations in Two Independent Variables

The Painlevé Test and Reducibility to the Canonical Forms for Higher-Dimensional Soliton Equations with Variable-Coefficients

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