Space–time memory functions and solution of nonlinear evolution equations

A new approach is presented for solving a certain class of nonlinear partial differential equations. A space–time memory function Λ(r,t) is introduced to exactly convert a given nonlinear evolution equation into the following linear form: (∂/∂t) f(r,t)=Ω(r) f(r,t) +∫t0dt′∫dr′Λ(r−r′,t−t′)f(r′,t′). A Markovian integro-differential operator Ω(r) and the memory function Λ(r,t) reflect the nonlinearity, and are determined depending on a given initial condition. The approach is useful if higher-order memory functions associated with Λ are insensitive to approximation. The Korteweg–de Vries equation is treated as an example. For certain initial profiles the memory function is shown to be identically zero, and we find exact linear partial differential equations leading to the single- and the two-soliton solution. In the case of the three-soliton solution, the second-order memory function vanishes exactly, and Λ(r,t) is found to be a single exponential function of t.