Contents Prefab Chapter I The solitary waves and solitons in shallow water trough §1 Waves, water waves and dispersion……………1 N2 The discovery of solitary waves…………………7 §3 The interaction of solitons……………… N4 Conservation laws… 17 85 The inverse scattering method for solving the initial-value problems of KdV equation .... 19 86 The experiments on the KdV solitons in water trough…………26 §7 Applications of KdⅤ equation and its generalization…28 Chapter 2 The envelope solitons in deep water trough………………………30 &8 The stokes wave on deep water and the side- band instability…………30 89 NLS equation and the intability of Stokes waves…·…33 8 10 The initial-value problems of