Quasi self-adjointness of a class of third order nonlinear dispersive equations

Torrisi, Mariano; Tracinà, Rita

doi:10.1016/j.nonrwa.2012.10.013

Cited by 38 publications

(24 citation statements)

References 24 publications

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“…We show that the set of adjoint symmetries admitted by the PDEs is identical to the one of differential substitutions of nonlinear self-adjointness, and then express the correspondence between symmetries, adjoint symmetries and conservation laws via formula (9), which avoids integral operation by multiplier method. Furthermore, we demonstrate that the set of differential substitution of nonlinear self-adjointness contains the one of conservation law multipliers as a subset.…”

Section: Resultsmentioning

confidence: 99%

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Conservation laws of partial differential equations: Symmetry, adjoint symmetry and nonlinear self-adjointness

Zhang

2017

Computers & Mathematics with Applications

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Nonlinear self-adjointness method for constructing conservation laws of partial differential equations (PDEs) is further studied. We show that any adjoint symmetry of PDEs is a differential substitution of nonlinear self-adjointness and vice versa. Consequently, each symmetry of PDEs corresponds to a conservation law via a formula if the system of PDEs is nonlinearly self-adjoint with differential substitution. As a byproduct, we find that the set of differential substitutions includes the set of conservation law multipliers as a subset. The results are illustrated by three typical examples.

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Section: Resultsmentioning

confidence: 99%

“…Recently, Ibragimov provides a special method, named by nonlinear self-adjointness method, to construct some conservation laws of PDEs [8][9][10]. The two required conditions of this approach are the admitted symmetries and the differential substitutions which convert nonlocal conservation laws to local ones.…”

Section: Introductionmentioning

confidence: 99%

Conservation laws of partial differential equations: Symmetry, adjoint symmetry and nonlinear self-adjointness

Zhang

2017

Computers & Mathematics with Applications

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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%

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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“…where the conserved densities C t (18) and C x (18) for equation (18) is given by (29) where v is the substitution for nonlinear self-adjointness of Eq. (18). The first-order approximate infinitesimal operator X = X 0 + ǫX…”

Section: Approximate Conservation Lawmentioning

confidence: 99%

Determination of approximate non-linear self-adjointness and approximate conservation law

Zhang

2014

IMA Journal of Applied Mathematics

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Approximate nonlinear self-adjointness is an effective method to construct approximate conservation law of perturbed partial differential equations (PDEs). In this paper, we study the relations between approximate nonlinear self-adjointness of perturbed PDEs and nonlinear self-adjointness of the corresponding unperturbed PDEs, and consequently provide a simple approach to discriminate approximate nonlinear self-adjointness of perturbed PDEs. Moreover, a succinct approximate conservation law formula by virtue of the known conservation law of the unperturbed PDEs is given in an explicit form. As an application, we classify a class of perturbed wave equations to be approximate nonlinear self-adjointness and construct the general approximate conservation laws formulae. The specific examples demonstrate that approximate nonlinear self-adjointness can generate new approximate conservation laws.

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Quasi self-adjointness of a class of third order nonlinear dispersive equations

Cited by 38 publications

References 24 publications

Conservation laws of partial differential equations: Symmetry, adjoint symmetry and nonlinear self-adjointness

Conservation laws of partial differential equations: Symmetry, adjoint symmetry and nonlinear self-adjointness

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

Determination of approximate non-linear self-adjointness and approximate conservation law

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