New conservation laws for inviscid Burgers equation

Freire, Igor Leite

doi:10.1590/s1807-03022012000300007

Cited by 22 publications

(15 citation statements)

References 18 publications

(31 reference statements)

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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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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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“…This shows that (7) is quasi self-adjoint admitting an arbitrary nonlinear substitution = ( ). For further details and discussion, see [19,20].…”

Section: Preliminariesmentioning

confidence: 99%

“…In [19], a general class of first order (1 + 1) PDE was classified with respect to strictly and quasi self-adjointness. Later, in [20], the subclass of the Riemman, or inviscid Burgers equation, was reconsidered from the point of view of nonlinear self-adjointness. Recently, in the paper [21], the last class was studied incorporating damping and conservation laws were established.…”

Section: Introductionmentioning

confidence: 99%

Nonlinear Self-Adjoint Classification of a Burgers-KdV Family of Equations

Sampaio¹,

Freire

2014

Abstract and Applied Analysis

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“…Among them, Burgers' equation may be the more intriguing one which serves as a prototype model for turbulent flows [3]. Many works in the literature are devoted to investigation of Lie symmetries and the exact solutions of Burgers' equation(s) and its generalizations [4][5][6][7][8][9][10][11][12][13][14]. Beyond its importance in mathematical physics, it appears also as a geodesic equation on diffeomorphism group of a circle with right invariant 2 metric (see [15] and references therein).…”

Section: Introductionmentioning

confidence: 99%

Burgers’ Equations in the Riemannian Geometry Associated with First-Order Differential Equations

Bayrakdar

2018

Advances in Mathematical Physics

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We construct metric connection associated with a first-order differential equation by means of the generator set of a Pfaffian system on a submanifold of an appropriate first-order jet bundle. We firstly show that the inviscid and viscous Burgers' equations describe surfaces attached to an ODE of the form / = ( , ) with certain Gaussian curvatures. In the case of PDEs, we show that the scalar curvature of a three-dimensional manifold encoding a system of first-order PDEs is determined in terms of the integrability condition and the Gaussian curvatures of the surfaces corresponding to the integral curves of the vector fields which are annihilated by the contact form. We see that an integral manifold of any PDE defines intrinsically flat and totally geodesic submanifold.

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New conservation laws for inviscid Burgers equation

Abstract: Abstract. In this paper it is shown that the inviscid Burgers equation is nonlinearly self-adjoint.Then, from Ibragimov's theorem on conservation laws, local conserved quantities are obtained.Mathematical subject classification: Primary: 76M60; Secondary: 58J70.

Cited by 22 publications

References 18 publications

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

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

Nonlinear Self-Adjoint Classification of a Burgers-KdV Family of Equations

Burgers’ Equations in the Riemannian Geometry Associated with First-Order Differential Equations

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