Some constructive applications of Λ 2 -representations to integration of PDEs

A. Popov; S. Zadadaev

Annales Polonici Mathematici (2000)

  • Volume: 74, Issue: 1, page 261-274
  • ISSN: 0066-2216

Abstract

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Two new applications of Λ 2 -representations of PDEs are presented: 1. Geometric algorithms for numerical integration of PDEs by constructing planimetric discrete nets on the Lobachevsky plane Λ 2 . 2. Employing Λ 2 -representations for the spectral-evolutionary problem for nonlinear PDEs within the inverse scattering problem method.

How to cite

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Popov, A., and Zadadaev, S.. "Some constructive applications of $Λ^{2}$-representations to integration of PDEs." Annales Polonici Mathematici 74.1 (2000): 261-274. <http://eudml.org/doc/208370>.

@article{Popov2000,
abstract = {Two new applications of $Λ^\{2\}$-representations of PDEs are presented: 1. Geometric algorithms for numerical integration of PDEs by constructing planimetric discrete nets on the Lobachevsky plane $Λ^\{2\}$. 2. Employing $Λ^\{2\}$-representations for the spectral-evolutionary problem for nonlinear PDEs within the inverse scattering problem method.},
author = {Popov, A., Zadadaev, S.},
journal = {Annales Polonici Mathematici},
keywords = {discrete net; $Λ^\{2\}$-representations of PDEs; Lobachevsky (hyperbolic) geometry; pseudospherical metric; geometrical algorithms; inverse scattering method; sine-Gordon equation},
language = {eng},
number = {1},
pages = {261-274},
title = {Some constructive applications of $Λ^\{2\}$-representations to integration of PDEs},
url = {http://eudml.org/doc/208370},
volume = {74},
year = {2000},
}

TY - JOUR
AU - Popov, A.
AU - Zadadaev, S.
TI - Some constructive applications of $Λ^{2}$-representations to integration of PDEs
JO - Annales Polonici Mathematici
PY - 2000
VL - 74
IS - 1
SP - 261
EP - 274
AB - Two new applications of $Λ^{2}$-representations of PDEs are presented: 1. Geometric algorithms for numerical integration of PDEs by constructing planimetric discrete nets on the Lobachevsky plane $Λ^{2}$. 2. Employing $Λ^{2}$-representations for the spectral-evolutionary problem for nonlinear PDEs within the inverse scattering problem method.
LA - eng
KW - discrete net; $Λ^{2}$-representations of PDEs; Lobachevsky (hyperbolic) geometry; pseudospherical metric; geometrical algorithms; inverse scattering method; sine-Gordon equation
UR - http://eudml.org/doc/208370
ER -

References

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  1. [1] M. J. Ablowitz and H. Segur, Solitons and the Inverse Scattering Transform, SIAM, Philadelphia, 1981. Zbl0472.35002
  2. [2] E. Cartan, Les systèmes différentiels extérieurs et leurs applications géométriques, Hermann, Paris, 1945. Zbl0211.12701
  3. [3] A. G. Popov, The non-Euclidean geometry and differential equations, in: Banach Center Publ. 33, Inst. Math., Polish Acad. Sci., 1996, 297-308. Zbl0851.35119
  4. [4] E. G. Poznyak and A. G. Popov, Lobachevsky geometry and the equations of mathematical physics, Russian Acad. Sci. Dokl. Math. 48 (1994), 338-342. Zbl0817.35098
  5. [5] E. G. Poznyak and A. G. Popov, Non-Euclidean geometry: Gauss formula and PDE's interpretation, Itogi Nauki i Tekhniki (VINITI), Geometry 2 (1994), 5-24 (in Russian). 
  6. [6] E. G. Poznyak and A. G. Popov, Geometry of the sine-Gordon equation, Itogi Nauki i Tekhniki (VINITI), Problems of Geometry, 23 (1991), 99-130 (in Russian). Zbl0741.35072
  7. [7] E. G. Poznyak and A. G. Popov, The Sine-Gordon Equation: Geometry and Physics, Znanie, Moscow, 1991 (in Russian). Zbl0790.53002
  8. [8] E. G. Poznyak and E. V. Shikin, Differential Geometry, Moscow Univ. Press, Moscow, 1990 (in Russian). 
  9. [9] A. A. Samarskĭ, Theory of Difference Schemes, Nauka, Moscow, 1977 (in Russian). 
  10. [10] R. Sasaki, Soliton equations and pseudospherical surfaces, Nuclear Phys. B 154 (1979), 343-357. 
  11. [11] A. S. Smogorzhevskiĭ, Geometric Constructions on the Lobachevsky Plane, Gostekhteorizdat, Moscow, 1951 (in Russian). 
  12. [12] L. A. Takhtadzhyan and L. D. Faddeev, Hamiltonian Approach in Soliton Theory, Nauka, Moscow, 1986 (in Russian). Zbl0632.58003
  13. [13] S. A. Zadadaev, Λ 2 -representations of equations of mathematical physics and formulation of the spectral-evolutionary problem, Vestnik Moskov. Univ. Fiz. Astronom. 1998, no. 5, 29-32 (in Russian). 
  14. [14] V. E. Zakharov and L. D. Faddeev, Korteweg-de Vries equation is a completely integrable Hamiltonian system, Funktsional. Anal. i Prilozhen. 5 (1971), no. 4, 18-127 (in Russian). 

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