Second harmonic generation: Hamiltonian structures and particular solutions

Abstract

Articles you may be interested inSecond harmonic generation in random composites of particles with core-shell structure A numerical study of optical second-harmonic generation in a one-dimensional photonic structure For the equations describing second harmonic generation ͑SHG͒, we find a linear system different from one found by Kaup ͑but connected with it by a gauge trans-formation͒, which enables us to consider the SHG equations as related to the first negative flow in the coupled KdV hierarchy and to find therefore the Hamiltonian structures and the recursion operator. With the help of the recursion operator we write down the simplest commuting flows and consider their stationary points in order to find particular solutions of the SHG equations. We also investigate some particular invariant solutions of the SHG equations that can be of physical interest, and find their connections with invariant solutions of the Tzitzéica equation. The latter are either connected with Painlevé III or solvable in elliptic functions.

The Gurevich–Zybin system

“…This is a good symptom that such equations should be linearizable. Since above mentioned linearizable reduction of the SHG system is determined by 2x2 spectral transform, but the SHG system is a some reduction of another important three-wave interaction problem (see for instance [20]), then we can assume such systems like the Gurevich-Zybin system are linearizable if they are in intersection of at least two different integrable hierarchies (e.g. the Liouville equation is a member of two different hierarchies: of the KdV equation and of the Sawada-Kotera equation).…”

“…In one-dimensional case, for short pulses, when the group-velocity mismatch between both frequency components becomes important, then the process of second harmonic generation (SHG) is described by the complex equations (see for instance [19], [20]), which in real form are (see [20], formula (6) therein)ũ…”

On soliton-type solutions of equations associated with N-component systems

“…Using asymptotics of these soliton solutions they also demonstrated soliton fission. A connection between the SHG system and the cKdV system was recently discussed by Khusnutdinova and Steudel(1998).…”

Second harmonic generation: Goursat problem on the semi-strip, Weyl functions and explicit solutions

“…The SHG integrability was proved and Lax pair was constructed in [12]. In spite of many important results (see [2,14,16,37] and references therein) SHG has remained unsolved. One the reasons for this situation is connected with the continuously interacting nature of SHG [15].…”

“…Bäcklund transformation is a well-known and fruitful approach in the nonlinear equations and spectral theory (see, for instance, [1,20,21,22,39]). Bäcklund transformations and construction of some self-similar solutions for SHG have been studied in [16,38,40]. GBDT have been initially developed in [25,26] (see [10,28,30,31] and references therein for various applications) and provides algebraic representation of the Darboux matrix in the form of the transform matrix from the system theory, which proves useful in the study of the explicit SHG solutions.…”