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Differential equations in the spectral parameter and multiphase similarity solutions

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Part of the Lecture Notes in Mathematics book series (LNM,volume 1324)

Abstract

Multiphase similarity solutions were introduced by Flaschka and Newell [1] by imposing a specific type of differential equation in the spectral parameter for solutions of linear systems. The most general Schroedinger equation such that its solutions satisfy a very general kind of differential equation in the spectral parameter has been found in [2].

Here we draw attention to some relations between the results in these papers by writing the equations in [2] in the form

rational and by showing that all solutions in [2] have the multiphase structure introduced in [1].

We also produce families of rational multiphase similarity solutions of the m KdV equation.

Keywords

  • Spectral Parameter
  • Rational Solution
  • General Kind
  • Isomonodromic Deformation
  • Multiphase Structure

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Partially supported by NSF Grant #DMS84-03232 and ONR Contract #N00014-84-C-0159

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References

  1. H. Flaschka, A. Newell, Multiphase similarity solutions of integrable evolution equations. Physica 3D (1981) 1 & 2, 203–221.

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  2. J.J. Duistermaat, F.A. Grünbaum, Differential equations in the spectral parameter. Comm. Math. Physics, 103, 177–240 (1986).

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  3. H. Flaschka, A commutator representation of Painlevé equation. J. Math. Physics 21(5), 1016–1018, 1980.

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  4. H. Flaschka, A. Newell, Monodromy-and spectrum-preserving deformations. Comm. Math. Physics 76, 65–116, (1980).

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  5. H. Airault, Rational solutions of Painlevé equations in App. Math. 61, 31–53 (1979).

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  6. M. Adler, J. Moser, On a class of polynomials connected with the KdV equation. Comm. in Math. Physics 61, 1–30 (1978)

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© 1988 Springer-Verlag

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Grünbaum, F.A. (1988). Differential equations in the spectral parameter and multiphase similarity solutions. In: Cardoso, F., de Figueiredo, D.G., Iório, R., Lopes, O. (eds) Partial Differential Equations. Lecture Notes in Mathematics, vol 1324. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0100786

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  • DOI: https://doi.org/10.1007/BFb0100786

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-50111-4

  • Online ISBN: 978-3-540-45928-6

  • eBook Packages: Springer Book Archive