## Solitons in Mathematics and PhysicsThe soliton is a dramatic concept in nonlinear science. What makes this book unique in the treatment of this subject is its focus on the properties that make the soliton physically ubiquitous and the soliton equation mathematically miraculous. Here, on the classical level, is the entity field theorists have been postulating for years: a local traveling wave pulse; a lump-like coherent structure; the solution of a field equation with remarkable stability and particle-like properties. It is a fundamental mode of propagation in gravity- driven surface and internal waves; in atmospheric waves; in ion acoustic and Langmuir waves in plasmas; in some laser waves in nonlinear media; and in many biologic contexts, such as alpha- helix proteins. |

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### Contents

Chapter | 1 |

Chapter 2 | 23 |

Chapter 3 | 61 |

Chapter 4 | 113 |

Connecting Links Among the Miracles of Soliton Mathematics | 145 |

5f Isospectral isoRiemann surfaces and isomonodromic deforma | 181 |

237 | |

### Common terms and phrases

ad-invariant AKNS hierarchy amplitude asymptotic expansion Backlund transformation behavior calculation Chapter coefficients commutator component conservation laws conserved density constant coordinates corresponding defined dependence derivatives discuss element exactly example exercise exponential finite flows formulae function fundamental solution matrix given gives Hamiltonian structure Hamiltonian vector field Hirota equations independent variables infinite dimensional infinite number initial integrability condition introduce inverse scattering Kac-Moody algebra KdV equation KdV family lattice Lax equations left factor loop algebra mass flux Mathematical monodromy motion multisoliton solutions N-soliton solution NLS equation nonlinear Schrodinger Note obtain ordinary differential equations Painleve Painleve property parameter perturbation phase shift phase space Poisson bracket polynomial pulses r-function reader recall Riemann right going satisfies scattering data Schlesinger transformation Schrodinger equation sine-Gordon equation solitary wave soliton equations solve subalgebras symmetries theory Toda lattice two-soliton velocity write zero