Nonclassical Symmetries and the Singular Manifold Method: Theory and Six Examples

DOI10.1002/sapm199595173zbMath0840.35003OpenAlexW2529871365MaRDI QIDQ4857049

Pilar Ruiz Gordoa, Pilar G. Estevez

Publication date: 11 July 1996

Published in: Studies in Applied Mathematics (Search for Journal in Brave)

Full work available at URL: https://doi.org/10.1002/sapm199595173

zbMATH Keywords

integrabilitysingular manifold method

Mathematics Subject Classification ID

KdV equations (Korteweg-de Vries equations) (35Q53) Invariance and symmetry properties for PDEs on manifolds (58J70) Other special methods applied to PDEs (35A25)

Related Items (max. 100)

Nonclassical symmetries as special solutions of heir-equations ⋮ New invariant sets to nonlinear diffusion equations with \(x\)-dependent convection and absorption ⋮ A KdV equation in 2+1 dimensions: Painlevé analysis, solutions and similarity reductions ⋮ Symbolic computation of exact solutions expressible in hyperbolic and elliptic functions for nonlinear PDEs ⋮ A fourth-order numerical scheme for solving the modified Burgers equation ⋮ Bounded traveling waves of the Burgers-Huxley equation ⋮ Symmetry reductions for a dissipation-modified KdV equation ⋮ Non-Isospectral 1+1 Hierarchies Arising from a Camassa Holm Hierarchy in 2+1 Dimensions ⋮ Extended rotation and scaling groups for nonlinear diffusion equations ⋮ Singular sector of the Kadomtsev–Petviashvili hierarchy, ∂̄ operators of nonzero index, and associated integrable systems ⋮ Persistence of travelling wave solutions of a fourth order diffusion system ⋮ Symmetry analysis and solutions for a family of Cahn-Hilliard equations ⋮ Extended rotation and scaling groups for nonlinear evolution equations ⋮ Nonclassical symmetries for a family of Cahn-Hilliard equations ⋮ Solution of moving boundary space-time fractional Burger's equation ⋮ Nonclassical symmetry reductions of the Boussinesq equation ⋮ Unified approach to Miura, Bäcklund and Darboux Transformations for Nonlinear Partial Differential Equations ⋮ A precise definition of reduction of partial differential equations ⋮ Analytic solutions of fractal and fractional time derivative-Burgers-Nagumo equation

Cites Work

• Unnamed Item
• Aspects of the Painlevé property for partial differential equations
• Similarity reductions from extended Painlevé expansions for nonintegrable evolution equations
• Painlevé expansions for nonintegrable evolution equations
• Symmetry reductions and exact solutions of a class of nonlinear heat equations
• The construction of special solutions to partial differential equations
• Nonlinear Evolution Equations Generated from the Backlund Transformation for the Boussinesq Equation
• Lie transformations, nonlinear evolution equations, and Painlevé forms
• Painleve analysis of the generalized Burgers-Huxley equation
• The Painlevé property for partial differential equations. II: Bäcklund transformation, Lax pairs, and the Schwarzian derivative
• Exact similarity solutions to some nonlinear diffusion equations
• Similarity reductions of the KP equation by a direct method
• New similarity reductions of the Boussinesq equation
• New similarity solutions for the modified Boussinesq equation
• Solitary wave solutions of the generalised Burgers-Huxley equation
• The Painlevé property for partial differential equations
• The Connection between Partial Differential Equations Soluble by Inverse Scattering and Ordinary Differential Equations of Painlevé Type
• Group-Invariant Solutions of Differential Equations
• The Painlevé-Kowalevski and Poly-Painlevé Tests for Integrability
• Similarity reductions of partial differential equations
• Nonclassical symmetry solutions and the methods of Bluman–Cole and Clarkson–Kruskal
• Direct reduction and differential constraints
• Algorithms for the Nonclassical Method of Symmetry Reductions
• Modified singular manifold expansion: application to the Boussinesq and Mikhailov-Shabat systems

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