Group classification of variable coefficient generalized Kawahara equations

Kuriksha, Oksana; Pošta, Severin; Vaneeva, Olena

doi:10.1088/1751-8113/47/4/045201

Cited by 17 publications

(22 citation statements)

References 61 publications

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“…and the authors of [15] did the same earlier for the special case when f (u) = u n . While equation (3) is slightly more general than (2), the generalized symmetries of either (2) or (3) to the best of our knowledge were not studied in full generality in the earlier literature.…”

Section: Resultsmentioning

confidence: 75%

Symmetries and conservation laws for a generalization of Kawahara equation

Vašíček

2020

Journal of Geometry and Physics

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We give a complete classification of generalized and formal symmetries and local conservation laws for a nonlinear evolution equation which generalizes the Kawahara equation having important applications in the study of plasma waves and capillary-gravity water waves.In particular, we show that the equation under study admits no genuinely generalized symmetries and has only finitely many local conservation laws, and thus this equation is not symmetry integrable.

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Section: Resultsmentioning

confidence: 75%

Symmetries and conservation laws for a generalization of Kawahara equation

Vašíček

2020

Journal of Geometry and Physics

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“…The set of admissible transformations of a class of differential equations naturally possesses the groupoid structure with respect to the standard operation of transformations composition [12]. More details and examples on finding and usage of admissible transformations for generalized fKdV equations as well as definitions of different kinds of equivalence groups can be found in [6,15].…”

Section: Admissible Transformationsmentioning

confidence: 99%

“…The presence of variable coefficients in a differential equation that model certain real-world phenomenon often allows one to get better description of the phenomenon but, at the same time, makes the related studies of this equation, including group classification problems, more difficult. In recent works on Lie symmetry analysis it was shown that the usage of admissible transformations in many cases is a cornerstone that leads to exhaustive solution of group classification problems [1,6,12,13]. That's why we firstly investigate admissible transformations in the class (1) in the next section and then proceed with the classification of Lie symmetries in Section 3.…”

Section: Introductionmentioning

confidence: 99%

Group Analysis of Generalized Fifth-Order Korteweg–de Vries Equations with Time-Dependent Coefficients

Kuriksha

Pošta

Vaneeva

2014

Springer Proceedings in Mathematics &Amp; Statistics

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“…Proposition 7 and Section 9. Variable-coefficient generalizations of the Kawahara equation were studied in a similar way in [17,18]. The group classification of Galilei-invariant equations of the form u t + uu x = F (u r ) was carried out in [5,8].…”

Section: Introductionmentioning

confidence: 99%

Group analysis of general Burgers–Korteweg–de Vries equations

Opanasenko

Bihlo

Popovych

2017

Journal of Mathematical Physics

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The complete group classification problem for the class of (1+1)-dimensional rth order general variable-coefficient Burgers-Korteweg-de Vries equations is solved for arbitrary values of r greater than or equal to two. We find the equivalence groupoids of this class and its various subclasses obtained by gauging equation coefficients with equivalence transformations. Showing that this class and certain gauged subclasses are normalized in the usual sense, we reduce the complete group classification problem for the entire class to that for the selected maximally gauged subclass, and it is the latter problem that is solved efficiently using the algebraic method of group classification. Similar studies are carried out for the two subclasses of equations with coefficients depending at most on the time or space variable, respectively. Applying an original technique, we classify Lie reductions of equations from the class under consideration with respect to its equivalence group. Studying alternative gauges for equation coefficients with equivalence transformations allows us not only to justify the choice of the most appropriate gauge for the group classification but also to construct for the first time classes of differential equations with nontrivial generalized equivalence group such that equivalence-transformation components corresponding to equation variables locally depend on nonconstant arbitrary elements of the class. For the subclass of equations with coefficients depending at most on the time variable, which is normalized in the extended generalized sense, we explicitly construct its extended generalized equivalence group in a rigorous way. The new notion of effective generalized equivalence group is introduced.

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Group classification of variable coefficient generalized Kawahara equations

Cited by 17 publications

References 61 publications

Symmetries and conservation laws for a generalization of Kawahara equation

Symmetries and conservation laws for a generalization of Kawahara equation

Group Analysis of Generalized Fifth-Order Korteweg–de Vries Equations with Time-Dependent Coefficients

Group analysis of general Burgers–Korteweg–de Vries equations

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