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 5 Preface......Page 7 1. The discovery of solitary waves......Page 9 2. Fundamental ideas......Page 12 3. The discovery of soliton interactions......Page 15 4. Applications of the KdV equation......Page 16 Problems......Page 17 2. Solitary waves......Page 21 3. General waves of permanent form......Page 22 4. Description of waves in terms of elliptic functions......Page 26 5. Infinitesimal waves......Page 28 6. Solitary waves again......Page 29 Problems......Page 31 1. Fundamental ideas......Page 40 2. Gardner's transformation......Page 42 Problems......Page 45 1. The problem......Page 48 2. Sketch of the method of inverse scattering......Page 49 3. The scattering problem......Page 50 4. The evolution equation......Page 53 5. Solution of the scattering problem for t > 0......Page 56 6. The inverse scattering problem......Page 58 7. Qualitative character of the solution......Page 59 8. Example: the delta-function potential......Page 61 9. Example: g(x) = - 2sech2x......Page 64 10. Example: g(x) = - 6sech2x......Page 67 11. Examples: sech-squared potentials......Page 70 12. Examples: some numerical results......Page 71 13. Reflectionless potentials......Page 75 Problems......Page 79 1. Description of the method in terms of operators......Page 87 Problems......Page 91 1. Introduction......Page 95 2. Waves and solitons......Page 97 3. Some other simple explicit solutions......Page 101 4. The interaction of two solitons......Page 102 5. A breather......Page 106 6. The method of inverse scattering......Page 107 Problems......Page 109 1. Introduction......Page 116 2. The sine-Gordon equation......Page 119 3. The KdV equation......Page 121 Problems......Page 126 1. Epilogue......Page 129 Appendix A DERIVATION OF THE INTEGRAL EQUATION FOR INVERSE SCATTERING......Page 133 Bibliography and author index......Page 136 Motion picture index......Page 141 Subject index......Page 142