[Paper Review] Equivalence groupoid of a class of general Burgers-Korteweg-de Vries equations with space-dependent coefficients

Stanislav Opanasenko|arXiv (Cornell University)|Aug 30, 2019

Nonlinear Waves and Solitons9 references4 citations

TL;DR

This paper classifies admissible transformations and equivalence groups for a class of generalized Burgers–Korteweg–de Vries equations with space-dependent coefficients. It shows that the reduced class of equations has a four-dimensional usual equivalence group and identifies several subclasses admitting maximal nontrivial conditional equivalence groups, including new examples of normalization in the generalized sense.

ABSTRACT

We describe the equivalence groupoid of the class of general Burgers - Korteweg - de Vries equations with space-dependent coefficients. This class is shown to reduce by a family of equivalence transformations to a subclass whose usual equivalence group is four-dimensional. Classified are admissible transformations of this subclass and singled out are its subclasses admitting maximal nontrivial conditional equivalence groups. All of them turn out to have dimension higher than four. In particular, a few new examples of nontrivial cases of normalization in the generalized sense of classes of differential equations appeared this way.

Motivation & Objective

• To analyze the equivalence groupoid of a class of general Burgers–Korteweg–de Vries equations with space-dependent coefficients.
• To reduce the class to a normalized subclass with a four-dimensional usual equivalence group.
• To classify admissible transformations and identify subclasses admitting maximal nontrivial conditional equivalence groups.
• To discover new examples of normalization in the generalized sense for classes of differential equations.
• To establish the structure of the equivalence groupoid via the usual equivalence group and generalized equivalence groups of normalized subclasses.

Proposed method

• The paper reduces the original class of equations to a subclass with $ C=1 $ and $ A^1=0 $, denoted $ ilde{ ho} $, which is shown to have a four-dimensional usual equivalence group.
• It classifies admissible transformations within the reduced class $ ilde{ ho} $, identifying specific subclasses with enhanced symmetry structures.
• The analysis employs point transformations and equivalence transformations to determine conditional equivalence groups, particularly focusing on generalized and effective generalized equivalence groups.
• The paper constructs explicit forms of equivalence transformations mapping equations in $ ilde{ ho} $ to equations in normalized subclasses, including those with power, logarithmic, exponential, and rational coefficient structures.
• It uses the concept of effective generalized equivalence groups to identify maximal nontrivial conditional equivalence subgroups for normalized subclasses.
• The equivalence groupoid of the reduced class is generated by its usual equivalence group and the equivalence groups of the identified normalized subclasses.

Experimental results

Research questions

• RQ1 What is the structure of the equivalence groupoid for the class of generalized Burgers–Korteweg–de Vries equations with space-dependent coefficients?
• RQ2 Which subclasses of the reduced equation class admit maximal nontrivial conditional equivalence groups?
• RQ3 Can new examples of normalization in the generalized sense be found within this class of equations?
• RQ4 How do equivalence transformations map equations from the reduced class to subclasses with enhanced symmetry?
• RQ5 What is the role of the usual equivalence group in generating the full equivalence groupoid of the reduced class?

Key findings

• The usual equivalence group of the reduced class $ ilde{ ho} $ of generalized Burgers–Korteweg–de Vries equations is four-dimensional.
• The equivalence groupoid of $ ilde{ ho} $ is generated by its usual equivalence group and the equivalence groups of seven normalized subclasses: $ ilde{ ho}_{ ext{I},1} $, $ ilde{ ho}_{ ext{I},01} $, $ ilde{ ho}_{ ext{I},00} $, $ ilde{ ho}_{ ext{II},1} $, $ ilde{ ho}_{ ext{III}} $, $ ilde{ ho}_{ ext{IV},1} $, $ ilde{ ho}_{ ext{IV},0}^{r>2} $, and $ ilde{ ho}_{ ext{IV},0}^{r=2} $.
• All subclasses except $ ilde{ ho}_{ ext{II},0} $ are normalized in the generalized sense, with their effective generalized equivalence groups serving as maximal conditional equivalence groups.
• The subclass $ ilde{ ho}_{ ext{II},0} $ is normalized in the usual sense, with its usual equivalence group being four-dimensional.
• The complement class $ ilde{ ho}_0 $, defined as the union of all equations not in the above subclasses, is normalized in the usual sense with an explicitly described equivalence group.
• Explicit point transformations are derived that map equations from $ ilde{ ho}_{ ext{IV},00}^{r=2} $ to $ ilde{ ho}_{ ext{IV},0}^{r=2} $, with forms involving functions $ P^1, P^2, R^1, R^2 $, and parameters satisfying $ c_4 eq 0 $, $ ho eq 0 $, and $ P^2_{ar{t}} > 0 $.