On integrable conservation laws

Author:

Arsie Alessandro1,Lorenzoni Paolo2,Moro Antonio3

Affiliation:

1. Department of Mathematics and Statistics, University of Toledo, 2801 W. Bancroft St., Toledo, OH 43606, USA

2. Dipartimento di Matematica e Applicazioni, Università di Milano-Bicocca, Via Roberto Cozzi 53, 20125 Milano, Italy

3. Department of Mathematics and Information Sciences, University of Northumbria at Newcastle, Pandon Building, Camden St., Newcastle upon Tyne NE2 1XE, UK

Abstract

We study normal forms of scalar integrable dispersive (not necessarily Hamiltonian) conservation laws, via the Dubrovin–Zhang perturbative scheme. Our computations support the conjecture that such normal forms are parametrized by infinitely many arbitrary functions that can be identified with the coefficients of the quasi-linear part of the equation. Moreover, in general, we conjecture that two scalar integrable evolutionary partial differential equations having the same quasi-linear part are Miura equivalent. This conjecture is also consistent with the tensorial behaviour of these coefficients under general Miura transformations.

Publisher

The Royal Society

Subject

General Physics and Astronomy,General Engineering,General Mathematics

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