Nonlinear systems of the Burgers-type equations: Lie and Q-conditional symmetries, Ansätze and solutions

Cherniha, Roman; Serov, Mykola

doi:10.1016/s0022-247x(03)00155-0

Cited by 37 publications

(33 citation statements)

References 12 publications

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“…After the imposed vorticity decays, the state will again evolve according to the Schrödinger equation. Symmetry classification of some Burgers type systems is carried out in [41] (higher-order symmetries) and [42] (Lie and conditional symmetries).…”

Section: Some Relevant Questions In Symmetry Analysismentioning

confidence: 99%

Classical and Quantum Burgers Fluids: A Challenge for Group Analysis

Broadbridge

2015

Symmetry

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Abstract:The most general second order irrotational vector field evolution equation is constructed, that can be transformed to a single equation for the Cole-Hopf potential. The exact solution to the radial Burgers equation, with constant mass influx through a spherical supply surface, is constructed. The complex linear Schrödinger equation is equivalent to an integrable system of two coupled real vector equations of Burgers type. The first velocity field is the particle current divided by particle probability density. The second vector field gives a complex valued correction to the velocity that results in the correct quantum mechanical correction to the kinetic energy density of the Madelung fluid. It is proposed how to use symmetry analysis to systematically search for other constrained potential systems that generate a closed system of vector component evolution equations with constraints other than irrotationality.

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Section: Some Relevant Questions In Symmetry Analysismentioning

confidence: 99%

Classical and Quantum Burgers Fluids: A Challenge for Group Analysis

Broadbridge

2015

Symmetry

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“…To the best of our knowledge, there are not many paper devoted to search of Q-conditional symmetries for the systems of PDEs [20][21][22][23][24]. One may easily check that Definition 2 was only used in all these papers.…”

Section: Remarkmentioning

confidence: 99%

Reaction-Diffusion Systems with Constant Diffusivities: Conditional Symmetries and Form-Preserving Transformations

Cherniha

Davydovych

2014

Springer Proceedings in Mathematics &Amp; Statistics

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Q-conditional symmetries (nonclassical symmetries) for a general class of two-component reaction-diffusion systems with constant diffusivities are studied. Using the recently introduced notion of Q-conditional symmetries of the first type (R. Cherniha J. Phys. A: Math. Theor., 2010. vol. 43., 405207), an exhaustive list of reaction-diffusion systems admitting such symmetry is derived. The formpreserving transformations for this class of systems are constructed and it is shown that this list contains only non-equivalent systems. The obtained symmetries permit to reduce the reaction-diffusion systems under study to two-dimensional systems of ordinary differential equations and to find exact solutions. As a non-trivial example, multiparameter families of exact solutions are explicitly constructed for two nonlinear reaction-diffusion systems. A possible interpretation to a biologically motivated model is presented.

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“…Since then the nonclassical symmetry method has been applied to various equations and systems in hundreds of published papers, e.g. [27], [37], [16], [28], [17], [54], [13], [11], [12], [15], [51], [4], the latest b being [14], [31], [55], [10].…”

Section: Introductionmentioning

confidence: 99%