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 Algebra i Analiz: Year: Volume: Issue: Page: Find

 Algebra i Analiz, 1994, Volume 6, Issue 3, Pages 40–58 (Mi aa450)

Research Papers

On some developments of the $\overline\partial$-dressing method

L. V. Bogdanov, V. E. Zakharov

Landau Institute for Theoretical Physics, Centre for Non-linear Studies

Abstract: Some developments of the $\overline\partial$-dressing method concerning an algebraic scheme of constructing integrable equations and construction of solutions with special properties are considered. It is demonstrated how the matrix $\mathbf{KP}$ equation appears from the scalar dressing and, more generally, how to construct the integrable system corresponding to an arbitrary triad of polynomials. Using the nonlocal $\overline\partial$-problem approach in $(2+1)$ dimensions, it is shown that the $\overline\partial$-problem with a shift and (for decreasing solutions) the Riemann problem with a shift naturally arise in $(1+1)$ dimensions. The Boussinesq equation and the first order relativistically-invariant systems are investigated. The developed approach allows one also to investigate the structure of the continuous spectrum and the inverse scattering problem for an arbitrary order ordinary differential operator on the infinite line.

Keywords: Inverse scattering problem method, the dressing method, $\bar\partial$-problem, the Riemann problem.

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English version:
St. Petersburg Mathematical Journal, 1995, 6:3, 475–493

Bibliographic databases:

Language: English

Citation: L. V. Bogdanov, V. E. Zakharov, “On some developments of the $\overline\partial$-dressing method”, Algebra i Analiz, 6:3 (1994), 40–58; St. Petersburg Math. J., 6:3 (1995), 475–493

Citation in format AMSBIB
\Bibitem{BogZak94} \by L.~V.~Bogdanov, V.~E.~Zakharov \paper On some developments of the $\overline\partial$-dressing method \jour Algebra i Analiz \yr 1994 \vol 6 \issue 3 \pages 40--58 \mathnet{http://mi.mathnet.ru/aa450} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=1301829} \zmath{https://zbmath.org/?q=an:0839.35115|0813.35102} \transl \jour St. Petersburg Math. J. \yr 1995 \vol 6 \issue 3 \pages 475--493 

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Erratum

This publication is cited in the following articles:
1. Konopelchenko, BG, “Singular sector of the Kadomtsev-Petviashvili hierarchy, (partial derivative)over-bar operators of nonzero index, and associated integrable systems”, Journal of Mathematical Physics, 41:1 (2000), 385
2. Bogdanov, L, “The Boussinesq equation revisited”, Physica D-Nonlinear Phenomena, 165:3–4 (2002), 137
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