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A new integrable nonlinear integrodifferential equation (NIDE) is proposed. This equation may be interpreted as a model equation for deep-water waves. The N-periodic and N-soliton solutions for the equation are constructed by means of the bilinear transformation method. These solutions have the same structure as that for the Benjamin-Ono equation which describes internal waves in stratified fluids of great depth. Furthermore, it is shown that the motion of the positions of the poles of solutions is related to certain solvable finite-dimensional dynamical systems described by first-order nonlinear ordinary differential equations. The discussion is also made on a more general NIDE that may be interpreted as a model equation describing nonlinear waves in fluids of finite depth. [ABSTRACT FROM AUTHOR]

A systematic method for constructing particular solutions of the nonlinear Klein–Gordon and Liouville equations in four-spatial dimensions is developed. The method of solution presented here first consists of reducing nonlinear partial differential equations to ordinary differential equations (ODE’s) by introducing symmetry variables and then seeking exact solutions for more tractable ODE’s. Various exact solutions are presented, in which new solutions with nonspherical symmetries are included. Furthermore, the exact method is applied to the above equations in general n-spatial dimensions. Among them, a conformally invariant nonlinear Klein–Gordon equation is particularly interesting from the viewpoint of field theories. The exact solutions for these equations are generalizations of those for the corresponding equations in four-spatial dimensions. [ABSTRACT FROM AUTHOR]