# Some constructive applications of ${\Lambda}^{2}$-representations to integration of PDEs

Annales Polonici Mathematici (2000)

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

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topPopov, 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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- [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] 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] 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] 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] E. G. Poznyak and A. G. Popov, The Sine-Gordon Equation: Geometry and Physics, Znanie, Moscow, 1991 (in Russian). Zbl0790.53002
- [8] E. G. Poznyak and E. V. Shikin, Differential Geometry, Moscow Univ. Press, Moscow, 1990 (in Russian).
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- [10] R. Sasaki, Soliton equations and pseudospherical surfaces, Nuclear Phys. B 154 (1979), 343-357.
- [11] A. S. Smogorzhevskiĭ, Geometric Constructions on the Lobachevsky Plane, Gostekhteorizdat, Moscow, 1951 (in Russian).
- [12] L. A. Takhtadzhyan and L. D. Faddeev, Hamiltonian Approach in Soliton Theory, Nauka, Moscow, 1986 (in Russian). Zbl0632.58003
- [13] S. A. Zadadaev, ${\Lambda}^{2}$-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] 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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