The Unified Transform Method for Linear PDEs

There exist certain nonlinear evolution PDEs which can be mapped to linear evolution PDEs. The prototypical such linearizable PDE is the so-called Burger's equation u t − u xx = 2uu x , (1.1) which can be written in the conservation form ∂ t u = ∂ x (u x + u 2). Employing the so-called Cole-Hopf transformation u = q x q (1.2) and using the identity ∂ t q x q = ∂ t ∂ x ln q = ∂ x ∂ t ln q = ∂ x q t q , we find, upon an x-integration, the heat equation q t − q xx = 0. Burger's equation arises in several applications involving nonlinear diffusive processes. A prototypical equation arising in several applications involving nonlin-ear dispersive processes is the celebrated Korteweg-deVries (KdV) equation u t + u x + u xxx + uu x = 0. (1.4) 1 1 INTRODUCTION 2 Is there a transformation analogous to (1.2) which maps (1.4) to the corresponding linear equation q t + q x + q xxx = 0, (1.5) called the Stokes equation? The answer to this question is negative. However, there does exist a more subtle linearization procedure, which is based on the existence of the so-called Lax pair formulation. Equations which admit such a formulation are called integrable PDEs. Such equations are rare, however many of them are physically significant (F. Calogero has presented a heuristic argument for the explanation of this fact). The simplest integrable nonlinear evolution PDE in one spatial dimension is the so-called nonlinear Schrödinger (NLS) equation iu t + u xx − 2σ|u| 2 u = 0, σ = ±1, u ∈ C.