Integrability and Self-Similarity in Transient Stimulated Raman Scattering

Summary. The phenomenon of stimulated Raman scattering (SRS) can be described by three coupled PDEs which define the pump electric field, the Stokes electric field, and the material excitation as functions of distance and time. In the transient limit these equations are integrable, i.e., they admit a Lax pair formulation. Here we study this transient limit. The relevant physical problem can be formulated as an initial-boundary value (IBV) problem where both independent variables are on a finite domain. A general method for solving IBV problems for integrable equations has been introduced recently. Using this method we show that the solution of the equations describing the transient SRS can be obtained by solving a certain linear integral equation. It is interesting that this equation is identical to the linear integral equation characterizing the solution of an IBV problem of the sine-Gordon equation in light-cone coordinates. This integral equation can be solved uniquely in terms of the values of the pump and Stokes fields at the entry of the Raman cell. The asymptotic analysis of this solution reveals that the long-distance behavior of the system is dominated by the underlying self-similar solution which satisfies a particular case of the third Painlevé transcendent. This result is consistent with both numerical simulations and experimental observations. We also discuss briefly the effect of frequency mismatch between the pump and the Stokes electric fields. Received December 10, 1996; second revision received October 10, 1997; final revision received January 20, 1998     

On the solvability of Painlevé II and IV

We introduce a rigorous methodology for studying the Riemann-Hilbert problems associated with certain integrable nonlinear ordinary differential equations. For concreteness we investigate the Painlevé II and Painlevé IV equations. We show that the Cauchy problems for these equations admit in general global, meromorphic int solutions. Furthermore, for special relations among the monodromy data and fort on Stokes lines, these solutions are bounded for finitet.     

Interaction of Simple Particles in Soliton Cellular Automata

We consider a certain cellular automaton recently introduced by Park, Steiglitz, and Thurston. By introducing appropriate mathematical notation, the interaction of simple particles evolving according to this automaton rule is completely characterized analytically. It is found that: (1) If two particles have different speed and they interact, then they interact solitonically and, although they may interact a number of times, they finally separate with the faster particle moving in front of the slower one. (2) If two particles have the same speed and are close enough so that they interact, there exist two cases: either they will interact only once and then they will separate, travelling independently of each other, or they will form a new periodic configuration by interacting forever.     

On the inverse scattering and direct linearizing transforms for the Kadomtsev-Petviashvili equation

The existing formalism for solving the initial-value problem associated with the Kadomtsev-Petviashvili equation, a physically significant two-spatial dimensional analogue of the Korteweg-de Vries equation, is both simplified and extended. The lump solutions, algebraically decaying solitons, are naturally incorporated and given a spectral characterization in the new scheme. In addition, a very general linearization of the Kadomtsev-Petviashvili equation is presented.     

Group theoretical aspects of constants of motion and separable solutions in classical mechanics

The aim of this paper is to establish the group nature of all separable solutions and conserved quantities in classical mechanics by analyzing the group structure of the Hamilton-Jacobi equation. It is shown that consideration of only classical Lie point groups is insufficient. For this purpose the Lie-Bäcklund groups of tangent transformations, rigorously established by Ibragimov and Anderson, are used. It is also shown how these generalized groups induce Lie groups on Hamilton's equations. The generalization of the above results to any order partial differential equation, where the dependent variable does not appear explicitly, is obvious. In the second part of the paper we investigate a certain class of admissible operators of the time-independent Hamilton-Jacobi equation of any energy state including the zero state. It is shown that in the latter case additional symmetries may appear. Finally, some potentials of physical interest admitting higher symmetries are considered.     

Symplectic structures,their Bäcklund transformations and hereditary symmetries

It is shown that compatible symplectic structures lead in a natural way to hereditary symmetries. (We recall that a hereditary symmetry is an operator-valued function which immediately yields a hierarchy of evolution equations, each having infinitely many commuting symmetries all generated by this hereditary symmetry. Furthermore this hereditary symmetry usually describes completely the soliton structure and the conservation laws of these equations). This result then provide us with a method for constructing hereditary symmetries and hence exactly solvable evolution equations.In addition we show how symplectic structures transform under Bäcklund transformations. This leads to a method for generating a whole class of symplectic structures from a given one.Several examples and applications are given illustrating the above results. Also the connection of our results with those of Gelfand and Dikii, and of Magri is briefly pointed out.     

On the Inverse Scattering Transform for the Kadomtsev-Petviashvili Equation

The initial value problem of the Kadomtsev-Petviashvili equation for one choice of sign in the equation has been recently investigated in the literature. Here we consider the other choice of sign. We introduce suitable eigenfunctions which though bounded are not analytic in the spectral parameter. This, in contrast to the known case, prevents us from formulating the inverse problem as a nonlocal Riemann-Hilbert boundary value problem. Nevertheless a suitable formulation is given and a formal solution is constructed via a linear integral equation.     

Boundary value problems for systems of linear evolution equations

It is shown that the new method for solving initial-boundaryvalue problems for scalar evolution equations recently introducedby one of the authors can also be applied to systems of evolutionequations. The novel step needed in this case is the constructionof a scalar Lax pair by using a suitable parametrization ofthe dispersion relation as well as certain linear transformations.The simultaneous spectral analysis of the Lax pair yields thesolution of a given initial-boundary value problem in termsof an integral in the complex spectral plane which involvesan appropriate x-transform of the initial conditions and anappropriate t-transform of the boundary conditions. These transformsare neither the x-Fourier transform nor the t-Laplace transform,rather they are new transforms custom made for the given systemof partial differential equations (PDEs) and the given domain.This method is illustrated by solving on the half-line the linearizedequations governing infinitesimal deformations in a heat conductingbar. Received 8 January 2003. Revised 18 July 2003. ^* Email: p.a.treharne{at}damtp.cam.ac.uk Email: t.fokas{at}damtp.cam.ac.uk