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Lectures on nonlinear integrable equations and their solutions · Around

2018

Lectures on nonlinear integrable equations and their solutions

Zabrodin, A.

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This is an introductory course on nonlinear integrable partial differential and differential-difference equ\-a\-ti\-ons based on lectures given for students of Moscow Institute of Physics and Technology and Higher School of Economics.

The typical examples of Korteweg-de Vries (KdV), Kadomtsev-Petviashvili (KP) and Toda lattice equations are studied in detail.
We give a detailed description of the Lax representation of these equations and their hierarchies in terms of pseudo-differential or pseudo-difference operators and also of different classes of the solutions including famous soliton solutions.
The formulation in terms of tau-function and Hirota bilinear differential and difference equations is also discussed.

Built on

I.M. Krichever, Elliptic solutions of the Kadomtsev-Petviashvili equation and integrable systems of particles
1980
Earlier work this paper cites.
B. Dubrovin, Theta functions and nonlinear equations
1981
Earlier work this paper cites.
E. Date, M. Kashiwara, M. Jimbo and T. Miwa, Transformation groups for soliton equations: Nonlinear integrable systems – classical theory and quantum theory
1983
Earlier work this paper cites.
M. Jimbo and T. Miwa, Solitons and infinite dimensional Lie algebras
1983
Earlier work this paper cites.

Similar

S. Novikov, S. Manakov, L. Pitaevskii and V. Zakharov, Theory of solitons: The inverse scattering method
1984
Cited alongside, same era.
K. Ueno and K. Takasaki, Toda lattice hierarchy
1984
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B. Dubrovin, I. Krichever and S. Novikov, Integrable systems I
1985
Cited alongside, same era.
A. Newell, Solitons in mathematics and physics
1985
Cited alongside, same era.

Then

L.A. Dickey, Soliton equations and Hamiltonian systems
1991
Later among the works it cites.
T. Miwa, M. Jimbo and E. Date, Solitons: differential equations, symmetries and infinite dimensional algebras
2000
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O. Babelon, D. Bernard, M. Talon, Introduction to classical integrable systems
2003
Later among the works it cites.
T. Takebe, Lectures on dispersionless integrable hierarchies
2014
Later among the works it cites.

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