Introduction to Multidimensional Integrable Equations

1. Front Matter

   Pages i-x

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2. Introduction

   • B. G. Konopelchenko

   Pages 1-45

3. The Inverse Spectral Transform Method in 2+1 Dimensions

   • B. G. Konopelchenko

   Pages 47-111

4. Other Integrable Equations and Methods of Solution in 2+1 Dimensions

   • B. G. Konopelchenko

   Pages 113-154

5. General Methods for the Construction of (2+1)-Dimensional Integrable Equations. τ-Function and ∂̄-Dressing Methods

   • B. G. Konopelchenko

   Pages 155-202

6. Multidimensional Integrable Systems

   • B. G. Konopelchenko

   Pages 203-236

7. Conclusion

   • B. G. Konopelchenko

   Pages 237-238

8. Back Matter

   Pages 239-292

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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.

• Soliton
• integrable system
• solution
• spectra

• Loughborough University of Technology, Leicestershire, England

  C. Rogers

• Institute of Nuclear Physics, Novosibirsk, Russia

  B. G. Konopelchenko

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