The Korteweg–deVries Equation: A Survey of Results

Abstract

The Korteweg–de Vries equation \[ u_t + uu_x + u_{xxx} = 0\] is a nonlinear partial differential equation arising in the study of a number of different physical systems, e.g., water waves, plasma physics, anharmonic lattices, and elastic rods. It describes the long time evolution of small-but-finite amplitude dispersive waves. From detailed studies of properties of the equation and its solutions, the concept of solitons was introduced and the method for exact solution of the initial-value problem using inverse scattering theory was developed. A survey of these and other results for the Korteweg–deVries equation are given, including conservation laws, an alternate method for exact solution, soliton solutions, asymptotic behavior of solutions, Bäcklund transformation, and a nonlinear WKB method. The recent literature contains many extensions of these ideas to a number of other nonlinear evolution equations of physical interest and to other classes of equations. Some of these equations and results are indicated. The paper concludes with a list of open problems.