Handbook of nonlinear partial differential equations.

*(English)*Zbl 1053.35001
Boca Raton, FL: Chapman & Hall/CRC (ISBN 1-58488-355-3/hbk). xx, 814 p. (2004).

The handbook under review contains all known exact solutions of nonlinear partial differential equations, which are exposed in chapters according to their type: parabolic, hyperbolic, elliptic, equations involving mixed derivatives, third order equations, equations of higher order. Each type of equations is classified according to its type of nonlinearity: equations with power-law nonlinearity, exponential, hyperbolic, logarithmic, trigonometric and arbitrary nonlinearities. Separately equations with two or more space variables are considered.

For equations in two independent variables and one unknown the following solution structure is presented: traveling-wave solution, additive separable solution, multiplicative separable, self-similar, generalized self-similar and traveling-wave, functional separable solutions.

Special attention is paid to concrete equations of mathematical physics. For parabolic type these are the nonlinear Schrödinger and related equations, for equations involving mixed derivatives these are the Calogero, Khokhlov-Zabolotskaya equations, equation of unsteady transonic gas flow, Monge-Ampère equation, for third order and fourth order equations the Korteweg-de Vries, Burgers-Korteweg-de Vries and Boussinesq, Kadomtsev-Petviashvili equations, respectively.

The book contains more than 1600 nonlinear partial differential equations. To each type of equation various applications are indicated. The book presents exact solutions to equations of heat and mass transfer, wave theory, nonlinear mechanics, hydro- and gas dynamics, nonlinear optics and acoustics, elasticity theory, flame and combustion theory, differential geometry, control theory, chemical engineering sciences and biology. Exact solutious of equations depending on arbitrary functions are important for testing numerical and approximate methods.

The supplemental chapters S1–S12 are devoted to short but sufficiently detailed presentations of exact methods for solving of nonlinear PDEs. Among them similarity method, generalized separation of variables and functional separation of variables methods, generalized similarity reductions of nonliniar equations, group analytic methods, inverse scattering and differential constraints methods, Painlevé test for nonlinear equations of mathematical physics, conservations laws and Noetherian symmetries are presented. Many exact solutions of nonlinear PDEs were obtained by the authors of this handbook.

This book will be very useful to pure and applied mathematicians, physicists, to all scientists and engineers, university teachers and students applying in their work nonlinear partial differential equations.

For equations in two independent variables and one unknown the following solution structure is presented: traveling-wave solution, additive separable solution, multiplicative separable, self-similar, generalized self-similar and traveling-wave, functional separable solutions.

Special attention is paid to concrete equations of mathematical physics. For parabolic type these are the nonlinear Schrödinger and related equations, for equations involving mixed derivatives these are the Calogero, Khokhlov-Zabolotskaya equations, equation of unsteady transonic gas flow, Monge-Ampère equation, for third order and fourth order equations the Korteweg-de Vries, Burgers-Korteweg-de Vries and Boussinesq, Kadomtsev-Petviashvili equations, respectively.

The book contains more than 1600 nonlinear partial differential equations. To each type of equation various applications are indicated. The book presents exact solutions to equations of heat and mass transfer, wave theory, nonlinear mechanics, hydro- and gas dynamics, nonlinear optics and acoustics, elasticity theory, flame and combustion theory, differential geometry, control theory, chemical engineering sciences and biology. Exact solutious of equations depending on arbitrary functions are important for testing numerical and approximate methods.

The supplemental chapters S1–S12 are devoted to short but sufficiently detailed presentations of exact methods for solving of nonlinear PDEs. Among them similarity method, generalized separation of variables and functional separation of variables methods, generalized similarity reductions of nonliniar equations, group analytic methods, inverse scattering and differential constraints methods, Painlevé test for nonlinear equations of mathematical physics, conservations laws and Noetherian symmetries are presented. Many exact solutions of nonlinear PDEs were obtained by the authors of this handbook.

This book will be very useful to pure and applied mathematicians, physicists, to all scientists and engineers, university teachers and students applying in their work nonlinear partial differential equations.

Reviewer: Boris V. Loginov (Ul’yanovsk)

##### MSC:

35-00 | General reference works (handbooks, dictionaries, bibliographies, etc.) pertaining to partial differential equations |

35C05 | Solutions to PDEs in closed form |

35A22 | Transform methods (e.g., integral transforms) applied to PDEs |

35A25 | Other special methods applied to PDEs |

35Qxx | Partial differential equations of mathematical physics and other areas of application |

35Jxx | Elliptic equations and elliptic systems |

35Kxx | Parabolic equations and parabolic systems |

35Lxx | Hyperbolic equations and hyperbolic systems |