Symmetries of a class of nonlinear third-order partial differential equations

Clarkson, Peter A.; Mansfield, Elizabeth L.; Priestley, Thomas J.

doi:10.1016/s0895-7177(97)00069-1

Cited by 52 publications

(31 citation statements)

References 54 publications

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“…To a strong solution u to (1) with initial data u 0 ∈ H 3 (R) and with maximal existence time T > 0 (given by Theorem A) we associate the differential equation (14) q t = u(t, q), t ∈ (0, T ), q(0, x) = x, x ∈ R.…”

Section: The Blow-up Setmentioning

confidence: 99%

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On the blow-up rate and the blow-up set of breaking waves for a shallow water equation

Escher

Constantin

2000

Math Z

304

140

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We consider the problem of the development of singularities for classical solutions to a new periodic shallow water equation. Blow-up can occur only in the form of wave-breaking, i.e. the solution remains bounded but its slope becomes unbounded in finite time. A quite detailed description of the wave-breaking phenomenon is given: there is at least a point (in general depending on time) where the slope becomes infinite exactly at breaking time. The precise blow-up rate is established and for a large class of initial data we also determine the blow-up set.Mathematics Subject Classification (1991): 35L05.

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Section: The Blow-up Setmentioning

confidence: 99%

“…As already mentioned in the Introduction, (14) is the equation for the geodesic in D 3 (S) issuing from the identity in the direction u 0 .…”

Section: The Blow-up Setmentioning

confidence: 99%

On the blow-up rate and the blow-up set of breaking waves for a shallow water equation

Escher

Constantin

2000

Math Z

304

140

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“…In [24] it was shown that the CH equation has also the same property, as well as in [4] it was proved that the Novikov equation is strictly self-adjoint. In particular, with respect to (3) and (4), the obtained results in [6], [24] and [4] shows some common facts:…”

Section: Historical Surveymentioning

confidence: 54%

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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“…In contrast the method of differential Gröbner bases (DGBs), which we describe below, has proved effective in coping with such difficulties (cf. [20,24,50,51]). …”

Section: U(x T) = a Exp (−|X − Ct|) mentioning

confidence: 99%

Symmetries of a Class of Nonlinear Fourth Order Partial Differential Equations

Clarkson¹,

Priestley²

1999

JNMP

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In this paper we study symmetry reductions of a class of nonlinear fourth order partial differential equationswhere α, β, γ, κ and µ are arbitrary constants. This equation may be thought of as a fourth order analogue of a generalization of the Camassa-Holm equation, about which there has been considerable recent interest. Further equation (1) is a "Boussinesqtype" equation which arises as a model of vibrations of an anharmonic mass-spring chain and admits both "compacton" and conventional solitons. A catalogue of symmetry reductions for equation (1) is obtained using the classical Lie method and the nonclassical method due to Bluman and Cole. In particular we obtain several reductions using the nonclassical method which are not obtainable through the classical method.

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Symmetries of a class of nonlinear third-order partial differential equations

Cited by 52 publications

References 54 publications

On the blow-up rate and the blow-up set of breaking waves for a shallow water equation

On the blow-up rate and the blow-up set of breaking waves for a shallow water equation

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

Symmetries of a Class of Nonlinear Fourth Order Partial Differential Equations

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