Group Classification of Linear Fourth-Order Evolution Equations

Huang, Qing; Qu, Changzheng; Zhdanov, Renat

doi:10.1016/s0034-4877(12)60049-4

Cited by 8 publications

(19 citation statements)

References 10 publications

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“…114-118] for r = 2 and in [3] for r > 2. (The last paper enhanced and extended results of [11, Section III] on r = 3 and of [13] on r = 4.) This is why it is appropriate to exclude linear equations from the present consideration.…”

Section: Introductionsupporting

confidence: 69%

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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Section: Introductionsupporting

confidence: 69%

Group analysis of general Burgers–Korteweg–de Vries equations

Opanasenko

Bihlo

Popovych

2017

Journal of Mathematical Physics

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“…Due to the same fact, Lie invariant solutions of equations from the subclass (5) as well as other solutions that are constructed by symmetry-based methods specific for linear equations can be classified up to G ∼ -equivalence, which is discussed in Section 6. It is worth mentioning the difference in the above version of the algebraic method to the classification technique employed in [19,20], which was proposed in [47] and applied therein to the group classification of a class of second-order nonlinear evolution equations. This technique is implicitly based on the normalization property of a class L| S of (systems of) differential equations to be classified, and its main steps are the following:…”

Section: Resultsmentioning

confidence: 99%

“…See also [7,23] for further developments and applications of the algebraic method. Different techniques within the framework of the algebraic method of group classification were used in [2,13,14,19,20,25,27,47].…”

Section: Algebraic Methods Of Group Classificationmentioning

confidence: 99%

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Group classification of linear evolution equations

Bihlo¹,

Popovych

2017

Journal of Mathematical Analysis and Applications

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The group classification problem for the class of (1+1)-dimensional linear rth order evolution equations is solved for arbitrary values of r > 2. It is shown that a related maximally gauged class of homogeneous linear evolution equations is uniformly semi-normalized with respect to linear superposition of solutions and hence the complete group classification can be obtained using the algebraic method. We also compute exact solutions for equations from the class under consideration using Lie reduction and its specific generalizations for linear equations.

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“…The same problem for n = 4 was undertaken in [4]. In a very recent work, an extension to n + 1-dimensional case for n > 2 of the group classification problem for linear evolution equations has been carried out in [5] where the authors solved the problem in full generality and compared their results with those of n = 3 of [1] (also for n = 4 of [4]) commented that there is a redundant case, where A(x) = a ∈ R, B(x) = 0 with generators ∂ t , ∂ x , t∂ t + 1/3(x − 2at)∂ x corresponding to N = 6 in [1]), that can be removed by an equivalence transformation (in fact a simple Galilei transformation (t, x, u) → (t, x + at, u) is sufficient to set a = 0) and an omission of a constraint on one of the coefficients of the generators. Above, in Table 1, we reproduce the group classification table with the necessary corrections made.…”

Section: Classification Of Linear Equationsmentioning

confidence: 99%

Linearizability for third order evolution equations

Basarab-Horwath

Güngör

2017

Journal of Mathematical Physics

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The problem of linearization for third order evolution equations is considered. Criteria for testing equations for linearity are presented. A class of linearizable equations depending on arbitrary functions is obtained by requiring presence of an infinite-dimensional symmetry group. Linearizing transformations for this class are found using symmetry structure and local conservation laws. A number of special cases as examples are discussed. Their transformation to equations within the same class by differential substitutions and connection with KdV and mKdV equations are also reviewed in this framework.Comment: 18 page

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Group Classification of Linear Fourth-Order Evolution Equations

Cited by 8 publications

References 10 publications

Group analysis of general Burgers–Korteweg–de Vries equations

Group analysis of general Burgers–Korteweg–de Vries equations

Group classification of linear evolution equations

Linearizability for third order evolution equations

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