Journal ArticleDOI
On a unified approach to transformations and elementary solutions of Painlevé equations
TLDR
In this article, an algorithmic method was developed for investigating the transformation properties of second-order equations of Painleve type, which utilizes the singularity structure of these equations, yields explicit transformations which relate solutions of the Painlesve equations II-VI, with different parameters.Abstract:
An algorithmic method is developed for investigating the transformation properties of second‐order equations of Painleve type. This method, which utilizes the singularity structure of these equations, yields explicit transformations which relate solutions of the Painleve equations II–VI, with different parameters. These transformations easily generate rational and other elementary solutions of the equations. The relationship between Painleve equations and certain new equations quadratic in the second derivative of Painleve type is also discussed.read more
Citations
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Book ChapterDOI
Geometry of 2D topological field theories
TL;DR: In this paper, the theory of equations of associativity describing geometry of moduli spaces of 2D topological field theories is studied, where WDVV equations and Frobenius manifolds are discussed.
Journal ArticleDOI
New similarity reductions of the Boussinesq equation
TL;DR: In this paper, some new similarity reductions of the Boussinesq equation, which arises in several physical applications including shallow water waves and also is of considerable mathematical interest because it is a soliton equation solvable by inverse scattering, are presented.
Journal ArticleDOI
Nonclassical symmetry reductions of the Boussinesq equation
TL;DR: In this paper, the authors compare and contrast the application of the Lie method of infinitesimals, the direct method due to Clarkson and Kruskal, and the non-classical one due to Bluman and Cole.
Journal ArticleDOI
Painlevé Classification of a Class of Differential Equations of the Second Order and Second Degree
Journal ArticleDOI
Painlevé equations: nonlinear special functions
TL;DR: In this paper, the authors discuss some of the remarkable properties which the Painleve equations possess including connection formulae, Backlund transformations associated discrete equations, and hierarchies of exact solutions.
References
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Journal ArticleDOI
Method for solving the Korteweg-deVries equation
TL;DR: In this paper, a method for solving the initial value problem of the Korteweg-deVries equation is presented which is applicable to initial data that approach a constant sufficiently rapidly as
Book
Similarity methods for differential equations
George W. Bluman,Julian D. Cole +1 more
TL;DR: In this paper, the authors define the notion of groups of transformations and prove that a one-parameter group essentially contains only one infinitesimal transformation and is determined by it.
Journal ArticleDOI
A connection between nonlinear evolution equations and ordinary differential equations of P‐type. II
TL;DR: The connection between nonlinear partial differential equations solvable by inverse scattering transforms and nonlinear ODEs of P-type (i.e., no movable critical points) is discussed in this article.