Solitons (London Mathematical Society Lecture Note Series, Series Number 85)

A 'soliton' is a localized nonlinear wave of permanent form which may interact strongly with other solitons so that when they separate after the interaction they regain their original forms. This textbook is an account of the theory of solitons and of the diverse applications of the theory to nonlinear systems arising in the physical sciences. The essence of the book is an introduction to the method of inverse scattering. Solitary waves, cnoidal waves, conservation laws, the initial-value problem for the Korteweg-de Vries equation, the Lax method, the sine-Gordon equation and Backlund transformations are treated. The book will be useful for research workers who wish to learn about solitons as well as graduate students in mathematics, physics and engineering. CONTENTS......Page p0005.djvu Preface......Page p0007.djvu 1. The discovery of solitary waves......Page p0009.djvu 2. Fundamental ideas......Page p0012.djvu 3. The discovery of soliton interactions......Page p0015.djvu 4. Applications of the KdV equation......Page p0016.djvu Problems......Page p0017.djvu 2. Solitary waves......Page p0021.djvu 3. General waves of permanent form......Page p0022.djvu 4. Description of waves in terms of elliptic functions......Page p0026.djvu 5. Infinitesimal waves......Page p0028.djvu 6. Solitary waves again......Page p0029.djvu Problems......Page p0031.djvu 1. Fundamental ideas......Page p0040.djvu 2. Gardner's transformation......Page p0042.djvu Problems......Page p0045.djvu 1. The problem......Page p0048.djvu 2. Sketch of the method of inverse scattering......Page p0049.djvu 3. The scattering problem......Page p0050.djvu 4. The evolution equation......Page p0053.djvu 5. Solution of the scattering problem for t > 0......Page p0056.djvu 6. The inverse scattering problem......Page p0058.djvu 7. Qualitative character of the solution......Page p0059.djvu 8. Example: the delta-function potential......Page p0061.djvu 9. Example: g(x) = - 2sech2x......Page p0064.djvu 10. Example: g(x) = - 6sech2x......Page p0067.djvu 11. Examples: sech-squared potentials......Page p0070.djvu 12. Examples: some numerical results......Page p0071.djvu 13. Reflectionless potentials......Page p0075.djvu Problems......Page p0079.djvu 1. Description of the method in terms of operators......Page p0087.djvu Problems......Page p0091.djvu 1. Introduction......Page p0095.djvu 2. Waves and solitons......Page p0097.djvu 3. Some other simple explicit solutions......Page p0101.djvu 4. The interaction of two solitons......Page p0102.djvu 5. A breather......Page p0106.djvu 6. The method of inverse scattering......Page p0107.djvu Problems......Page p0109.djvu 1. Introduction......Page p0116.djvu 2. The sine-Gordon equation......Page p0119.djvu 3. The KdV equation......Page p0121.djvu Problems......Page p0126.djvu 1. Epilogue......Page p0129.djvu Appendix A DERIVATION OF THE INTEGRAL EQUATION FOR INVERSE SCATTERING......Page p0133.djvu Bibliography and author index......Page p0136.djvu Motion picture index......Page p0141.djvu Subject index......Page p0142.djvu