Nonlinear self-adjointness of a generalized fifth-order KdV equation

Freire, Igor Leite; Sampaio, Júlio Cesar Santos

doi:10.1088/1751-8113/45/3/032001

Cited by 43 publications

(40 citation statements)

References 14 publications

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“…In general, while the adjoint-symmetry/symmetry formula (72) (and, hence, Ibragimov's theorem) looks very appealing, it has major drawbacks that in many examples [14][15][16][17][20][21][22]] the selection of a symmetry must be fitted to the form of the adjoint-symmetry to produce a non-trivial conservation law, and that in other examples [17][18][19] no non-trivial conservation laws are produced when only translation symmetries are available. More importantly, it is not (as is sometimes claimed) a generalization of Noether's theorem to non-variational DE systems.…”

Section: Propositionmentioning

confidence: 99%

“…However, in several papers [17][18][19], this formula sometimes is seen to produce only trivial conservation laws, and sometimes, the formula does not produce all admitted conservation laws. Furthermore, in a number of papers [14][15][16][17][20][21][22], the use of translation symmetries is mysteriously avoided, and other more complicated symmetries are used instead, without explanation.…”

Section: Introductionmentioning

confidence: 99%

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On the Incompleteness of Ibragimov’s Conservation Law Theorem and Its Equivalence to a Standard Formula Using Symmetries and Adjoint-Symmetries

Anco

2017

Symmetry

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A conservation law theorem stated by N. Ibragimov along with its subsequent extensions are shown to be a special case of a standard formula that uses a pair consisting of a symmetry and an adjoint-symmetry to produce a conservation law through a well-known Fréchet derivative identity. Furthermore, the connection of this formula (and of Ibragimov's theorem) to the standard action of symmetries on conservation laws is explained, which accounts for a number of major drawbacks that have appeared in recent work using the formula to generate conservation laws. In particular, the formula can generate trivial conservation laws and does not always yield all non-trivial conservation laws unless the symmetry action on the set of these conservation laws is transitive. It is emphasized that all local conservation laws for any given system of differential equations can be found instead by a general method using adjoint-symmetries. This general method is a kind of adjoint version of the standard Lie method to find all local symmetries and is completely algorithmic. The relationship between this method, Noether's theorem and the symmetry/adjoint-symmetry formula is discussed.

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Section: Propositionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

On the Incompleteness of Ibragimov’s Conservation Law Theorem and Its Equivalence to a Standard Formula Using Symmetries and Adjoint-Symmetries

Anco

2017

Symmetry

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“…Since Ibragimov's concepts on self-adjointness have been introduced, a considerable number of papers has been dealing with the problem of finding classes of differential equations with some self-adjoint property, see, for instance, [10,11,12,13,14,16,32].…”

Section: Historical Surveymentioning

confidence: 99%

“…The conserved vector (15) also provides a known conservation law for the class of Riemann equations (12), but only if b = −2.…”

Section: Strict Self-adjointness and Invariancementioning

confidence: 99%

On certain shallow water models, scaling invariance and strict self-adjointness

Silva

Freire

2015

Proceeding Series of the Brazilian Society of Computational and Applied Mathematics

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Resumo: In this work we establish conditions for a class of third order partial differential equations to be strictly self-adjoint and scale invariant. The obtained family of equations includes the Benjamin-Bona-Mahony, Camassa-Holm and Novikov equations. Using the strict selfadjointness and Ibragimov's conservation theorem, we establish some local conservation laws for some of the mentioned equations.Palavras-chave: Strict self-adjointness, Ibragimov's conservation theorem, conservation laws. Historical surveyDuring the last century, a sequence of papers, starting with [27], showed and enlightened many properties of the well known Korteweg -de Vries equation(1)Later, in [1], a new equation called Benjamin-Bona-Mahoney (BBM), given bywas derived as an "alternative" for the KdV. Although equation (2) was derived using the same formal justification for obtaining (1), the differences between both equations are greater than the fact that (1) is an evolution equation whereas (2) is not. In [1] the authors found three conserved quantities on the solutions of (2). Later in [30], those conservation laws obtained were proved to be the only three admitted by (2). This fact shows a dramatic difference between BBM and KdV since the last one admits an infinite number of conserved quantities [28].More recently, Camassa and Holm [5] using Hamiltonian methods derived the famous CamassaHolm (CH) equationThe last equation possesses remarkable properties such as peakon solutions and a bi-hamiltoninan structure, see [5], which implies in the existence of an infinite number of conserved quantities, just like the KdV equation [15,28,26].Since then, a considerable number of papers have been dedicated to derive third order nonevolutionary dispersive equations having similar properties as those known to KdV and CH equation. To cite a few number of examples, it was derived in [8] an integrable equation having peakon solutions with first order nonlinearities, while in [9] another integrable equation, combining linear dispersion such as the KdV equation and a nonlinear dispersion like the CH

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“…Nonlinearly self-adjointness have been focused by Freire and Sampaio in [6], where the authors determined a class of nonlinear self-adjoint equations of fifth-order. In [2] Bozhkov, Freire and Ibragimov showed that the nonlinear self-adjointness of the Novikov equation (further details, see [2] and references therein) implies in the strictly self-adjointness of that equation.…”

Section: Conservation Laws For Inviscid Burgers Equationmentioning

confidence: 99%