Equivalence groupoid of a class of variable coefficient Korteweg–de Vries equations

Vaneeva, Olena; Pošta, Severin

doi:10.1063/1.5004973

Cited by 17 publications

(11 citation statements)

References 27 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Therein, Lie symmetries of equations from these three subclasses were studied using an early version of the algebraic method of group classification. See also a modern treatment of these results in [57]. At the same time, the algebraic method of group classification was (implicitly or explicitly) used mostly for normalized classes whose equivalence algebras are infinite-dimensional; see [6,7,30,49,50] and references therein.…”

Section: Resultsmentioning

confidence: 99%

Extended symmetry analysis of generalized Burgers equations

Pocheketa

Popovych

2017

Journal of Mathematical Physics

View full text Add to dashboard Cite

Using advanced classification techniques, we carry out the extended symmetry analysis of the class of generalized Burgers equations of the form $u_t+uu_x+f(t,x)u_{xx}=0$. This enhances all the previous results on symmetries of these equations and includes the description of admissible transformations, Lie symmetries, Lie and nonclassical reductions, conservation laws, potential admissible transformations and potential symmetries. The study is based on the fact that the class is normalized, and its equivalence group is finite-dimensional.Comment: 31 pages, 2 tables, minor correction

show abstract

Section: Resultsmentioning

confidence: 99%

Extended symmetry analysis of generalized Burgers equations

Pocheketa

Popovych

2017

Journal of Mathematical Physics

View full text Add to dashboard Cite

show abstract

“…Such transformations are called form-preserving [26] or admissible [42] or allowed transformations [61]. The classifications for such transformations for various classes of PDEs were carried out, in particular, in [20,22,25,38,47,53,54,57]. Ordered triplets, consisting of the initial and target equations and the transformations linking them, together with the operation of composition of transformations have the groupoid structure.…”

Section: Equivalence Groupoidmentioning

confidence: 99%

Classification of reduction operators and exact solutions of variable coefficient Newell–Whitehead–Segel equations

Vaneeva

Boyko

Zhalij

et al. 2019

Journal of Mathematical Analysis and Applications

Self Cite

View full text Add to dashboard Cite

A class of the Newell-Whitehead-Segel equations (also known as generalized Fisher equations and Newell-Whitehead equations) is studied with Lie and "nonclassical" symmetry points of view. The classifications of Lie reduction operators and of regular nonclassical reduction operators are performed. The set of admissible transformations (the equivalence groupoid) of the class is described exhaustively. The criterion of reducibility of variable coefficient Newell-Whitehead-Segel equations to their constant coefficient counterparts is derived. Wide families of exact solutions for such variable coefficient equations are constructed.

show abstract

“…In [10,47] allowed transformations were computed for the class of variable-coefficient KdV equations of the form u t + f (t, x)uu x + g(t, x)u xxx = 0 with f g = 0 and were then used in [10] to carry out the group classification of this class; see [45] for a modern interpretation of these results. An attempt to reproduce these results for the class of variable-coefficient Burgers equations of the form u t + f (t, x)uu x + g(t, x)u xx = 0 with f g = 0 was made in [38] supposing that admissible transformations of this class are similar to admissible transformations of its thirdorder counterpart but in fact the structure of the corresponding equivalence groupoid is totally different from that in [10].…”

Section: Introductionmentioning

confidence: 99%

Group analysis of general Burgers–Korteweg–de Vries equations

Opanasenko

Bihlo

Popovych

2017

Journal of Mathematical Physics

View full text Add to dashboard Cite

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.

show abstract

Equivalence groupoid of a class of variable coefficient Korteweg–de Vries equations

Cited by 17 publications

References 27 publications

Extended symmetry analysis of generalized Burgers equations

Extended symmetry analysis of generalized Burgers equations

Classification of reduction operators and exact solutions of variable coefficient Newell–Whitehead–Segel equations

Group analysis of general Burgers–Korteweg–de Vries equations

Contact Info

Product

Resources

About