## Nonlinear Dispersive Equations: Existence and Stability of Solitary and Periodic Travelling Wave SolutionsThis book provides a self-contained presentation of classical and new methods for studying wave phenomena that are related to the existence and stability of solitary and periodic travelling wave solutions for nonlinear dispersive evolution equations. Simplicity, concrete examples, and applications are emphasized throughout in order to make the material easily accessible. The list of classical nonlinear dispersive equations studied includes Korteweg-de Vries, Benjamin-Ono, and Schrodinger equations. Many special Jacobian elliptic functions play a role in these examples. The author brings the reader to the forefront of knowledge about some aspects of the theory and motivates future developments in this fascinating and rapidly growing field. The book can be used as an instructive study guide as well as a reference by students and mature scientists interested in nonlinear wave phenomena. |

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### Contents

Basic Models | 17 |

Solitary and Periodic Travelling Wave Solutions | 25 |

Initial Value Problem | 49 |

Definition of Stability | 61 |

Orbital Stabilitythe Classical Method | 69 |

GrillakisShatahStrausss Stability Approach | 91 |

Existence and Stability of Solitary Waves for the GBO | 105 |

More about the ConcentrationCompactness Principle | 127 |

Sobolev Spaces of Periodic Type | 206 |

The Symmetric Decreasing Rearrangement | 207 |

The Jacobian Elliptic Functions | 208 |

Appendix B Operator Theory | 211 |

PseudoDifferential Operators and heir Spectrum T | 229 |

Spectrum of Linear Operators Associated to Solitary Waves | 231 |

SturmLiouville Theory | 237 |

Floquet Theory | 240 |

Instability of Solitary Wave Solutions | 137 |

Stability of Cnoidal Waves | 161 |

Appendix A Sobolev Spaces and Elliptic Functions | 201 |

Sobolev Spaces | 204 |

245 | |

255 | |

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### Common terms and phrases

Angulo associated basic Benjamin BO equation Bona bounded Chapter cnoidal wave solutions compact condition consider deﬁned deﬁnition denote differential equation eigenfunction eigenvalue elliptic function established example fact ﬁg ﬁnd ﬁnishes the proof ﬁnite ﬁrst ﬁxed ﬂow follows Fourier transform fundamental period G D(A G L2 given GKdV global Grillakis Hence Hilbert space Hilbert transform Hs(R implies inequality initial data instability integration interval Jacobian elliptic functions KdV equation Korteweg-de Vries equation Lemma linear operator mean zero minimizing sequence Moreover negative eigenvalue nonlinear evolution equations nonlinear Schrödinger equation norm obtain orbit periodic solutions periodic travelling wave Poisson Summation Theorem proof of Theorem properties prove pseudo-differential operator result satisﬁes satisfying self-adjoint operator Sobolev spaces solitary wave solutions speciﬁc spectral spectrum stability of solitary stability theory sufﬁciently Suppose travelling wave solutions unique well-posedness