Conditional invariance of the nonlinear wave equation

Fushchich, V. I.; Serov, Mykola

doi:10.1007/bf01670077

Cited by 30 publications

(38 citation statements)

References 3 publications

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“…Ovsiannikov [78] developed the method of partially invariant solutions. Bluman and Cole [9], in their study of symmetry reductions of the linear heat equation, proposed the so-called nonclassical method of group-invariant solutions (in the sequel referred to as the nonclassical method), which is also known as the "method of conditional symmetries" [33,34,36,37,59,76,81,82,103] and the "method of partial symmetries of the first type" [97]. In this method, the original pde (1.1) is augmented with the invariant surface condition ψ ≡ ξ(x, t, u)u x + τ (x, t, u)u t − φ(x, t, u) = 0, (1.4) which is associated with the vector field (1.3).…”

Section: ) Is Invariant Under This Transformation Yields An Overdetementioning

confidence: 99%

Symmetry reductions and exact solutions of a class of nonlinear heat equations

Clarkson

Mansfield

1994

Physica D: Nonlinear Phenomena

254

290

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Classical and nonclassical symmetries of the nonlinear heat equationare considered. The method of differential Gröbner bases is used both to find the conditions on f (u) under which symmetries other than the trivial spatial and temporal translational symmetries exist, and to solve the determining equations for the infinitesimals. A catalogue of symmetry reductions is given including some new reductions for the linear heat equation and a catalogue of exact solutions of (1) for cubic f (u) in terms of the roots of f (u) = 0. 0

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Section: ) Is Invariant Under This Transformation Yields An Overdetementioning

confidence: 99%

Symmetry reductions and exact solutions of a class of nonlinear heat equations

Clarkson

Mansfield

1994

Physica D: Nonlinear Phenomena

254

290

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“…In view of the constraintfũx = 0 for the class H, the equation (10) with z = x and the equation (12) respectively reduce to X xx = 0 and V u + V xux + u x V uux = 0. The expansion of the last equation implies U uu = 0.…”

Section: Equivalence Transformationsmentioning

confidence: 99%

Enhanced group classification of nonlinear diffusion–reaction equations with gradient-dependent diffusivity

Opanasenko

Boyko

Popovych

2020

Journal of Mathematical Analysis and Applications

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We carry out the enhanced group classification of a class of (1+1)-dimensional nonlinear diffusion-reaction equations with gradient-dependent diffusivity using the two-step version of the method of furcate splitting. For simultaneously finding the equivalence groups of an unnormalized class of differential equations and a collection of its subclasses, we suggest an optimized version of the direct method. The optimization includes the preliminary study of admissible transformations within the entire class and the successive splitting of the corresponding determining equations with respect to arbitrary elements and their derivatives depending on auxiliary constraints associated with each of required subclasses. In the course of applying the suggested technique to subclasses of the class under consideration, we construct, for the first time, a nontrivial example of finite-dimensional effective generalized equivalence group. Using the method of Lie reduction and the generalized separation of variables, exact solutions of some equations under consideration are found.

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“…In the "new" coordinates, the vector field (8.11) reads simply 18) and the (obviously X-invariant) asymptotic solution (8.7) is…”

Section: Solution-preserving Map Associated To Scalingmentioning

confidence: 99%

Asymptotic Scaling Symmetries for Nonlinear Pdes

Gaeta

Mancinelli

2005

Int. J. Geom. Methods Mod. Phys.

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Summary. In some cases, solutions to nonlinear PDEs happen to be asymptotically (for large x and/or t) invariant under a group G which is not a symmetry of the equation. After recalling the geometrical meaning of symmetries of differential equations -and solution-preserving maps -we provide a precise definition of asymptotic symmetries of PDEs; we deal in particular, for ease of discussion and physical relevance, with scaling and translation symmetries of scalar equations. We apply the general discussion to a class of "Richardson-like" anomalous diffusion and reaction-diffusion equations, whose solution are known by numerical experiments to be asymptotically scale invariant; we obtain an analytical explanation of the numerically observed asymptotic scaling properties. We also apply our method to a different class of anomalous diffusion equations, relevant in optical lattices. The methods developed here can be applied to more general equations, as clear by their geometrical construction.

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Conditional invariance of the nonlinear wave equation

Cited by 30 publications

References 3 publications

Symmetry reductions and exact solutions of a class of nonlinear heat equations

Symmetry reductions and exact solutions of a class of nonlinear heat equations

Enhanced group classification of nonlinear diffusion–reaction equations with gradient-dependent diffusivity

Asymptotic Scaling Symmetries for Nonlinear Pdes

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