The nonlinear Schrödinger equation. Self-focusing and wave collapse.

*(English)*Zbl 0928.35157
Applied Mathematical Sciences. 139. New York, NY: Springer. xvi, 350 p. (1999).

This is a fundamental monograph devoted to description of the wave collapse in models based on two- and three-dimensional fully or partly self-focusing nonlinear Schrödinger (NLS) equation (corresponding, respectively, to the self-focusing NLS equation with elliptic or hyperbolic transverse-diffraction/dispersion operator) and related systems of coupled equations. Both rigorous mathematical results and less rigorous “physical” ones are expounded in the book, along with results of numerical simulations.

The main topics presented in the book include the derivation of the NLS equation in various physical contexts, rigorous results concerning the existence and long-time asymptotic behavior of solutions localized in space, as well as stability of various (e.g., quasi-one-dimensional) solitons, and detailed asymptotic analysis of the blowing-up solutions that account for the collapse problem, i.e., formation of a singularity during a finite time. In the case of the strong collapse (which takes place, in the NLS equation in the three-dimensional space), the collapse is asymptotically described by self-similar solutions, while a more complicated analysis is necessary in the case of the weak collapse (which occurs at a boundary between the collapse-free and strongly collapsing models, e.g., for the NLS equation in the two-dimensional space).

Additionally, two large sections of the book are dealing with the collapse problem in models in which an NLS-like equation is coupled to an additional equation for a “mean field” (e.g., the Davey-Stewartson system) or to an equation for a field of the acoustic type (e.g., the Zakharov’s system).

The main topics presented in the book include the derivation of the NLS equation in various physical contexts, rigorous results concerning the existence and long-time asymptotic behavior of solutions localized in space, as well as stability of various (e.g., quasi-one-dimensional) solitons, and detailed asymptotic analysis of the blowing-up solutions that account for the collapse problem, i.e., formation of a singularity during a finite time. In the case of the strong collapse (which takes place, in the NLS equation in the three-dimensional space), the collapse is asymptotically described by self-similar solutions, while a more complicated analysis is necessary in the case of the weak collapse (which occurs at a boundary between the collapse-free and strongly collapsing models, e.g., for the NLS equation in the two-dimensional space).

Additionally, two large sections of the book are dealing with the collapse problem in models in which an NLS-like equation is coupled to an additional equation for a “mean field” (e.g., the Davey-Stewartson system) or to an equation for a field of the acoustic type (e.g., the Zakharov’s system).

Reviewer: B.A.Malomed (Tel Aviv)

##### MSC:

35Q55 | NLS equations (nonlinear Schrödinger equations) |

35-02 | Research exposition (monographs, survey articles) pertaining to partial differential equations |

37K40 | Soliton theory, asymptotic behavior of solutions of infinite-dimensional Hamiltonian systems |

37K45 | Stability problems for infinite-dimensional Hamiltonian and Lagrangian systems |

76B15 | Water waves, gravity waves; dispersion and scattering, nonlinear interaction |

76D33 | Waves for incompressible viscous fluids |

78A60 | Lasers, masers, optical bistability, nonlinear optics |

82D10 | Statistical mechanical studies of plasmas |