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Introduction to Multidimensional Integrable Equations : The Inverse Spectral Transform in 2+1 Dimensions - Boris Georgievich Konopelchenko

Introduction to Multidimensional Integrable Equations

The Inverse Spectral Transform in 2+1 Dimensions

Hardcover Published: 31st January 1993
ISBN: 9780306442209
Number Of Pages: 292

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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.

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

ISBN: 9780306442209
ISBN-10: 0306442205
Series: Language of Science
Audience: Professional
Format: Hardcover
Language: English
Number Of Pages: 292
Published: 31st January 1993
Country of Publication: US
Dimensions (cm): 23.39 x 15.6  x 1.91
Weight (kg): 0.61