Group foliation and non-invariant solutions of the heavenly equation

Martina, L.; Sheftel, M. B.; Winternitz, P.

doi:10.1088/0305-4470/34/43/310

Cited by 53 publications

(95 citation statements)

References 22 publications

(25 reference statements)

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“…In section 6, using results of the previous section, we obtain non-invariant solutions of HCMA by the lift from our non-invariant solutions to hyperbolic BF (1.3) that we obtained earlier [10]. Noninvariant solutions to the elliptic BF were obtained first by D. Calderbank and P. Tod [3].…”

Section: Introductionmentioning

confidence: 89%

Lift of Invariant to Non-Invariant Solutions of Complex Monge-Ampère Equations

Sheftel¹,

Malykh

2008

JNMP

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We show how partner symmetries of the elliptic and hyperbolic complex Monge-Ampère equations (CMA and HCMA) provide a lift of non-invariant solutions of three-and twodimensional reduced equations, i.e., a lift of invariant solutions of the original CMA and HCMA equations, to non-invariant solutions of the latter four-dimensional equations. The lift is applied to non-invariant solutions of the two-dimensional Helmholtz equation to yield non-invariant solutions of CMA, and to non-invariant solutions of three-dimensional wave equation and three-dimensional hyperbolic Boyer-Finley equation to yield non-invariant solutions of HCMA. By using these solutions as metric potentials, it may be possible to construct four-dimensional Ricci-flat metrics of Euclidean and ultra-hyperbolic signatures that have non-zero curvature tensors and no Killing vectors.

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Section: Introductionmentioning

confidence: 89%

Lift of Invariant to Non-Invariant Solutions of Complex Monge-Ampère Equations

Sheftel¹,

Malykh

2008

JNMP

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“…[11,12,13,14,15,16] when the group G of point symmetries is infinite-dimensional, and later it was developed in Ref. [6,7,8] when the point symmetry group G is finite-dimensional.…”

Section: Methods Of Group Foliationmentioning

confidence: 99%

Exact solutions of semilinear radial Schrödinger equations by separation of group foliation variables

Anco

Feng

Wolf

2015

Journal of Mathematical Analysis and Applications

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Abstract. Explicit solutions are obtained for a class of semilinear radial Schrödinger equations with power nonlinearities in multi-dimensions. These solutions include new similarity solutions and other new group-invariant solutions, as well as new solutions that are not invariant under any symmetries of this class of equations. Many of the solutions have interesting analytical behavior connected with blow-up and dispersion. Several interesting nonlinearity powers arise in these solutions, including the case of the critical (pseudo-conformal) power. In contrast, standard symmetry reduction methods lead to nonlinear ordinary differential equations for which few if any explicit solutions can be derived by standard integration methods.

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“…[10,11,33,[42][43][44][45][46][47]); -extend the applications to construct solutions from admitted symmetries to include admitted "symmetries" arising from generalizations (similarity solutions arising from the nonclassical and other related methods) (cf. [10,42,[48][49][50][51][52][53][54][55]); -efficiently solve the (overdetermined) linear systems of determining equations for symmetries or conservation law multipliers and solve the nonlinear systems of determining equations for the nonclassical and related methods through the development of symbolic manipulation software (cf. [42,[56][57][58][59][60][61][62][63][64][65][66][67][68][69][70][71]); -develop numerical schemes that effectively use symmetries and/or conservation laws for ODE's (cf.…”

Section: Similaritymentioning

confidence: 99%