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Setting Limits on Supersymmetry Using Simplified Models
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On integrable conservation laws.

Alessandro Arsie1, Paolo Lorenzoni2, Antonio Moro3

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

Proceedings. Mathematical, Physical, and Engineering Sciences
|January 9, 2015
PubMed
Summary
This summary is machine-generated.

We explore normal forms for integrable dispersive conservation laws using the Dubrovin-Zhang method. Our findings suggest these forms are linked to the equation's quasi-linear part, supporting a new Miura equivalence conjecture.

Keywords:
conservation lawsintegrabilitynormal forms

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Area of Science:

  • Mathematical Physics
  • Differential Equations
  • Nonlinear Dynamics

Background:

  • Scalar integrable dispersive conservation laws are crucial in modeling wave phenomena.
  • Understanding their normal forms is key to classifying and analyzing these equations.
  • The Dubrovin-Zhang perturbative scheme offers a method for this analysis.

Purpose of the Study:

  • To investigate the normal forms of scalar, integrable, dispersive (not necessarily Hamiltonian) conservation laws.
  • To test the conjecture that normal forms are parametrized by arbitrary functions related to the quasi-linear part.
  • To propose and support a general conjecture on Miura equivalence for such equations.

Main Methods:

  • Application of the Dubrovin-Zhang perturbative scheme.
  • Detailed computations of normal forms for specific integrable systems.
  • Analysis of the structure of coefficients under transformations.

Main Results:

  • Computations support the conjecture that normal forms are parametrized by infinitely many arbitrary functions.
  • These functions correspond to the coefficients of the quasi-linear part of the equation.
  • Evidence is found for the conjecture that equations with the same quasi-linear part are Miura equivalent.

Conclusions:

  • The study provides strong support for the parametrization of normal forms by quasi-linear coefficients.
  • A general conjecture on Miura equivalence is proposed and supported by computational and tensorial analysis.
  • This work advances the understanding of integrable systems and their classification.