Analytic study of a coupled Kerr-SBS system

doi:10.1016/j.cnsns.2016.05.008

Communications in Nonlinear Science and Numerical Simulation

Volume 42, January 2017, Pages 146-157

https://doi.org/10.1016/j.cnsns.2016.05.008 Get rights and content

Abstract

In order to describe the coupling between the Kerr nonlinearity and the stimulated Brillouin scattering, Mauger et al. recently proposed a system of partial differential equations in three complex amplitudes. We perform here its analytic study by two methods. The first method is to investigate the structure of singularities, in order to possibly find closed form single-valued solutions obeying this structure. The second method is to look at the infinitesimal symmetries of the system in order to build reductions to a lesser number of independent variables. Our overall conclusion is that the structure of singularities is too intricate to obtain closed form solutions by the usual methods. One of our results is the proof of the nonexistence of traveling waves.

Section snippets

The coupled Kerr-SBS system

The coupling between Kerr effect and stimulated Brillouin scattering [1] can be described by three complex partial differential equations (PDE) in three complex amplitudes U₁, U₂, Q depending on four independent variables x, y, t, z [8, (7), (9)] ${\begin{matrix} i (U_{1, z} + v_{g} U_{1, t}) + \frac{U_{1, x x} + U_{1, y y}}{2 k_{0}} + b (| {U_{1}}^{2} + 2 | {U_{2}}^{2}) U_{1} + i \frac{g}{2} Q U_{2} = 0, \\ - i (U_{2, z} - v_{g} U_{2, t}) + \frac{U_{2, x x} + U_{2, y y}}{2 k_{0}} + b (| {U_{2}}^{2} + 2 | {U_{1}}^{2}) U_{2} - i \frac{g}{2} \bar{Q} U_{1} = 0, \\ τ Q_{t} + Q - U_{1} {\bar{U}}_{2} = 0, \end{matrix}$ in which v_g, k₀, b, g, τ are real constants. We adopt the notation of nonlinear optics, in which the time t and the

Singularity analysis

There exists only one limiting case in which the system (1) is integrable, this is its degeneracy to the nonlinear Schrödinger equation $U_{1} = U_{2}, g = 0, \partial_{z} = 0, c_{1} \partial_{x} + c_{2} \partial_{y} = 0, (c_{1}, c_{2}) \neq (0, 0)$ . Let us prove that, except for this limiting case, the system (1) is always nonintegrable, in the sense that it always admits a multi-valued behaviour around a singularity which depends on the initial conditions (i.e. what is called a movable singularity). It is convenient to denote the list of dependent variables $(U_{1}, {\bar{U}}_{1}$

Search for radial shock-type solutions, generic case $g τ \neq 0$

For the radial reduction $\partial_{θ} = 0$ suggested by the local analysis, see (16), let us look for possible closed form singlevalued solutions defined by the assumption $\begin{matrix} u = \sum_{j = 0}^{1} u_{j} χ^{j - 1}, χ = φ - φ_{0}, \end{matrix}$ i.e. ${\begin{matrix} U_{1} = M e^{i a_{1}} (χ^{- 1} + U_{1, 1}), {\bar{U}}_{1} = M e^{- i a_{1}} (χ^{- 1} + {\bar{U}}_{1, 1}), \\ U_{2} = M e^{i a_{2}} (χ^{- 1} + U_{2, 1}), {\bar{U}}_{2} = M e^{- i a_{2}} (χ^{- 1} + {\bar{U}}_{2, 1}), \\ Q = N e^{i a_{1} - i a_{2}} (χ^{- 1} + Q_{1}), \bar{Q} = N e^{i a_{2} - i a_{1}} (χ^{- 1} + {\bar{Q}}_{1}), \end{matrix}$ in which the functions M, N, a₁, a₂, φ must obey the relations (4) and (7), and the functions U_{1, 1}, U_{2, 1}, Q₁ are to be determined. When one inserts such an assumption into the six

Lie symmetries

For convenience, we denote the independent variales x, y, z, t as $x_{j}, j = 1, 2, 3, 4$ and the dependent variables $U_{1}, {\bar{U}}_{1}, U_{2}, {\bar{U}}_{2}, Q, \bar{Q}$ as $u_{k}, k = 1, \dots, 6$ .

The method of Lie consists in unveiling the invariance properties of a given system of PDEs, in order to define reductions to another system with a lesser number of independent variables. We refer the reader to pedagogical textbooks such as [6], [10], [11], and to a recent paper [9] handling an example in full detail.

In order to apply the classical method

Reductions

Let us give a few examples of such reductions.

Conclusion

We have unveiled both the singularity structure and the underlying symmetries of a nonlinear optics system which has great potential applications. However, this analytic structure is too intricate to allow us to derive close form solutions. Since the original nonlinear system results from a reductive perturbation method, maybe another physical assumption during its derivation could make it tractable by the analytic techniques investigated here.

Acknowledgments

Both authors are happy to acknowledge the generous support of the Centro internacional de Ciencias in Cuernavaca. RC is grateful to Universidad de Cádiz for his visit, and thanks the organizers of the Workshop on laser-matter interaction (WLMI, Porquerolles, 2012) for their invitation.

References (12)

R. Conte et al.
Analytic expressions of hydrothermal waves
Rep Math Phys
(2000)
G.P. Agrawal
Nonlinear fiber optics
(2001)
R. Conte et al.
The Painlevé handbook
Russian translation Metod Penleve y ego prilozhenia (Regular and chaotic dynamics, Moscow, 2011)
(2008)
A.P. Fordy et al.
Nonlinear Schrödinger equations and simple Lie algebras
Commun Math Phys
(1983)
É. Goursat
Cours d’analyse mathématique
(1924)

There are more references available in the full text version of this article.

Cited by (1)

Symmetry analysis, exact solutions and conservation laws of 3D Zakharov-Kuznetsov (ZK) equation
2022, Journal of Interdisciplinary Mathematics

View full text

Research paperAnalytic study of a coupled Kerr-SBS system

Abstract

Section snippets

The coupled Kerr-SBS system

Singularity analysis

Search for radial shock-type solutions, generic case gτ≠0

Lie symmetries

Reductions

Conclusion

Acknowledgments

Rep Math Phys

Nonlinear fiber optics

The Painlevé handbook

Russian translation Metod Penleve y ego prilozhenia (Regular and chaotic dynamics, Moscow, 2011)

Nonlinear Schrödinger equations and simple Lie algebras

Commun Math Phys

Cours d’analyse mathématique

Research paper
Analytic study of a coupled Kerr-SBS system

Search for radial shock-type solutions, generic case $g τ \neq 0$