Introduction to Multidimensional Integrable Equations

The Inverse Spectral Transform in 2+1 Dimensions

By: B.G. Konopelchenko

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Hardcover | 31 January 1993

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308 Pages

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26.04 x 17.78 x 1.27

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The soliton represents one ofthe most important ofnonlinear phenomena in modern physics. It constitutes an essentially localizedentity with a set ofremarkable properties. Solitons are found in various areas of physics from gravitation and field theory, plasma physics, and nonlinear optics to solid state physics and hydrodynamics. Nonlinear equations which describe soliton phenomena are ubiquitous. Solitons and the equations which commonly describe them are also of great mathematical interest. Thus, the dis covery in 1967and subsequent development ofthe inversescattering transform method that provides the mathematical structure underlying soliton theory constitutes one of the most important developments in modern theoretical physics. The inversescattering transform method is now established as a very powerful tool in the investigation of nonlinear partial differential equations. The inverse scattering transform method, since its discoverysome two decades ago, has been applied to a great variety of nonlinear equations which arise in diverse fields of physics. These include ordinary differential equations, partial differential equations, integrodifferential, and differential-difference equations. The inverse scattering trans form method has allowed the investigation of these equations in a manner comparable to that of the Fourier method for linear equations.

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Non-Fiction Mathematics Calculus & Mathematical Analysis Functional Analysis & Transforms Science Physics Mathematical Physics Classical Mathematics Wave Mechanics including Vibration & Acoustics Engineering & Technology Technology in General Maths for Engineers

Science in General Maths for Scientists

Introduction	p. 1
The inverse spectral transform method in 1+1 dimensions. Brief history and examples of integrable equations	p. 1
Methods of solution for (1+1)-dimensional integrable equations	p. 10
Multidimensional generalizations	p. 21
Methods of solution for (2+1)-dimensional integrable systems. Summary	p. 35
The inverse spectral transform method in 2+1 dimensions	p. 47
The Kadomtsev-Petviashvili - I equation	p. 47
The Kadomtsev-Petviashvili - II equation. Generalized analytic functions	p. 59
Exact solutions of the Kadomtsev-Petviashvili equation	p. 67
The Davey-Stewartson - I equation	p. 76
The Davey-Stewartson - II equation	p. 86
The Veselov-Novikov (NVN-I[subscript +] equation	p. 91
The NVN-I and NVN-I[subscript 0] equations	p. 101
The Nizhnik (NVN-II) equation	p. 106
Other integrable equations and methods of solution in 2+1 dimensions	p. 113
The multidimensional resonantly-interacting three-wave model	p. 113
The Ishimori equation. The Hirota method	p. 116
The Manakov-Zakharov-Mikhailov equation	p. 121
Nonlocal, cylindrical, and other generalizations of the Kadomtsev-Petviashvili equation	p. 130
The Mel'nikov system	p. 134
The modified Kadomtsev-Petviashvili and Gardner equations. The Miura transformation and gauge invariance	p. 140
Further integrable equations in 2+1 dimensions	p. 144
General methods for the construction of (2+1)-dimensional integrable equations. [Actual symbol not reproducible]-function and [actual symbol not reproducible]-dressing methods	p. 155
The [actual symbol not reproducible]-function, vertex operator, and infinite-dimensional groups for the KP hierarchy	p. 156
Generalization of the dressing method	p. 167
The general [actual symbol not reproducible]-dressing method	p. 172
The [actual symbol not reproducible]-dressing method with variable normalization	p. 184
Operator representation of the multidimensional integrable equations	p. 192
Multidimensional integrable systems	p. 203
The self-dual Yang-Mills equation	p. 203
The supersymmetric Yang-Mills equation	p. 213
Multidimensional integrable generalizations of the wave, sine-Gordon, and self-dual equations	p. 218
Obstacles to multidimensionalization of the inverse spectral transform method. I. The Born approximation	p. 226
Obstacles to multidimensionalization of the inverse spectral transform method. II. Nonlinear characterization of the inverse scattering data	p. 232
Conclusion	p. 237
References	p. 239
Index	p. 291
Table of Contents provided by Blackwell. All Rights Reserved.

Introduction to Multidimensional Integrable Equations

The Inverse Spectral Transform in 2+1 Dimensions

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