Reductions of the Benney equations

Gibbons, John; Tsarëv, S. P.

doi:10.1016/0375-9601(95)00954-x

Cited by 175 publications

(251 citation statements)

References 6 publications

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“…i = j, which was first derived in [17,18] in the theory of hydrodynamic reductions of Benney's moment equations, see also [21,29,30] for further discussion. For any solution µ i , v of the system (16) one can reconstruct λ i and w by virtue of (13).…”

Section: Examplesmentioning

confidence: 99%

“…Solutions of this type, known as nonlinear interactions of n planar simple waves, were extensively investigated in gas dynamics and magnetohydrodynamics in a series of publications [5,6,7,36,37,9,20]. Later, they appeared in the context of the dispersionless KP hierarchy [15,16,17,18,21,31,29,30] and the theory of integrable hydrodynamictype chains [34,35]. We will call a multidimensional system integrable if it possesses 'sufficiently many' n-component reductions of the form (2) for arbitrary n (the precise definition follows).…”

Section: Introductionmentioning

confidence: 99%

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On the Integrability of (2+1)-Dimensional Quasilinear Systems

Ferapontov

Хуснутдинова

2004

Commun. Math. Phys.

177

295

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A (2+1)-dimensional quasilinear system is said to be 'integrable' if it can be decoupled in infinitely many ways into a pair of compatible n-component onedimensional systems in Riemann invariants. Exact solutions described by these reductions, known as nonlinear interactions of planar simple waves, can be viewed as natural dispersionless analogs of n-gap solutions. It is demonstrated that the requirement of the existence of 'sufficiently many' n-component reductions provides the effective classification criterion. As an example of this approach we classify integrable (2+1)-dimensional systems of conservation laws possessing a convex quadratic entropy.MSC: 35L40, 35L65, 37K10.

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Section: Examplesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

On the Integrability of (2+1)-Dimensional Quasilinear Systems

Ferapontov

Хуснутдинова

2004

Commun. Math. Phys.

177

295

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“…However we will now consider families of distribution functions f , which are parameterised by finitely many N Riemann invariantsλ i (x, t). We are interested in the case [9], [10] where the function λ(p, x, t) is such that only N of the moments are independent. Then there are N characteristic speeds, assumed real and distinct, and N corresponding Riemann invariants (p i ,λ i ), so Benney's equations reduce to a diagonal system of hydrodynamic type with finitely many dependent variablesλ i ,…”

Section: Reductions Of the Moment Equationsmentioning

confidence: 99%

“…The construction of a more general family of solutions for equations of this type was outlined in [9] and [10]. Instead of considering the principal value integral (7), we now define a new function λ + (x, p, t) :…”

Section: Reductions Of the Moment Equationsmentioning

confidence: 99%

Hyperelliptic reduction of the Benney moment equations

Baldwin¹,

Gibbons²

2003

J. Phys. A: Math. Gen.

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Abstract. We consider N -parameter reductions of the Benney moment equations. These were shown in Gibbons and Tsarev (1996 Phys. Lett. A 211 19, 1999 Phys. Lett. A 258 263 ) to correspond to N −parameter families of conformal maps and to satisfy a particular system of PDE. A specific known example of this, the (N = 2) elliptic reduction (L Yu and J Gibbons 2000 Inverse Problems 16 605 ) is described. We then consider an analogous reduction for a genus 2 hyperelliptic curve (N = 3). The mapping function is given by the inversion of a 2nd kind Abelian integral on the Θ−divisor. This is found explicitly following a method given by Enolskii, Pronine and Richter (2003 J. Nonlinear Science 13 157).

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“…A general theory of such systems was developed in the papers [1,2,3]. This theory is based on the existence of sufficiently many of the hydrodynamic reductions [4,1] which has been proposed as the definition of integrability. In the first nontrivial case n = 2 the complete set of integrability conditions has been found in the paper [2] in the form of a complicated system of PDEs for the entries of the matrices A and B.…”

Section: Introductionmentioning

confidence: 99%

A family of (2+1)-dimensional hydrodynamic type systems possessing a pseudopotential

Odesskii

2008

Sel. math., New ser.

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We construct a family of integrable hydrodynamic type systems with three independent and n ≥ 2 dependent variables in terms of solutions of linear system of PDEs with rational coefficients. We choose the existence of a pseudopotential as a criterion of integrability. In the case n = 2 this family is a general solution of the classification problem for such systems. We give also an elliptic analog of this family in the case n > 2.

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Reductions of the Benney equations

Cited by 175 publications

References 6 publications

On the Integrability of (2+1)-Dimensional Quasilinear Systems

On the Integrability of (2+1)-Dimensional Quasilinear Systems

Hyperelliptic reduction of the Benney moment equations

A family of (2+1)-dimensional hydrodynamic type systems possessing a pseudopotential

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