Coupled Nonlinear Schrodinger Research Articles

Nonfocusing Instabilities in Coupled, Integrable Nonlinear Schrödinger pdes

instabilities that exist independently of the well-known modulational instability of the focusing NLS equation. The focusing versus defocusing behavior of scalar NLS fields is a well-known model for the corresponding behavior of pulse transmission in optical fibers in the anomalous (focusing) versus normal (defocusing) dispersion regime [19], [20]. For fibers with birefringence (induced by an asymmetry in the cross section), the scalar NLS fields for two orthogonal polarization modes couple nonlinearly [26]. Experiments by Rothenberg [32], [33] have demonstrated a new type of modulational instability in a birefringent normal dispersion fiber, and he proposes this cross-phase coupling instability as a mechanism for the generation of ultrafast, terahertz optical oscillations. In this paper the nonfocusing plane wave instability in an integrable coupled nonlinear Schrodinger (CNLS) partial differential equation system is contrasted with the focusing instability from two perspectives: traditional linearized stability analysis and integrable methods based on periodic inverse spectral theory. The latter approach is a crucial first step toward a nonlinear , nonlocal understanding of this new optical instability analogous to that developed for the focusing modulational instability of the sine-Gordon equations by Ercolani, Forest, and McLaughlin [13], [14], [15], [17] and the scalar NLS equation by Tracy, Chen, and Lee [36], [37], Forest and Lee [18], and McLaughlin, Li, and Overman [23], [24].

The stationary equations of a coupled nonlinear Schrödinger system

The integrable coupled nonlinear Schrödinger (CNLS) equations under periodic boundary conditions are known to possess linearized instabilities in both the focussing and defocussing cases [M.G. Forest, D.W. McLaughlin, D. Muraki, O.C. Wright, Non-focussing instabilities in coupled, Integrable nonlinear Schrödinger PDEs, in preparation; D.J. Muraki, O.C. Wright, D.W. McLaughlin, Birefringent optical fibers: Modulational instability in a near-integrable system, Nonlinear Processes in Physics: Proceedings of III Postdam-V Kiev Workshop, 1991, pp. 242–245; O.C. Wright, Modulational stability in a defocussing coupled nonlinear Schrödinger system, Physica D 82 (1995) 1–10], whereas the scalar NLS equation is linearly unstable only in the focussing case [M.G. Forest, J.E. Lee, Geometry and modulation theory for the periodic Schrödinger equation, in: Dafermas et al. (Eds.), Oscillation Theory, Computation, and Methods of Compensated Compactness, I.M.A. Math. Appl. 2 (1986) 35–70]. These instabilities indicate the presence of crossed homoclinic orbits similar to those in the phase plane of the unforced Duffing oscillator [Y. Li, D.W. McLaughlin, Morse and Melnikov functions for NLS pde’s, Commun. Math. Phys. 162 (1994) 175–214; D.W. McLaughlin, E.A. Overman, Whiskered tori for integrable Pde’s: Chaotic behaviour in near integrable Pde’s, in: Keller et al. (Eds.), Surveys in Applied Mathematics, vol. 1, Chapter 2, Plenum Press, New York, 1995]. The homoclinic orbits and the near homoclinic tori that are connected to the unstable wave trains of the NLS and the CNLS reside in the finite-dimensional phase space of certain stationary equations [S.P. Novikov, Funct. Anal. Prilozen, 8 (3) (1974) 54–66] of the infinite hierarchy of integrable commuting flows. The correct stationary equations must be matched to the unstable torus through the analytic structure of the spectral curves [O.C. Wright, Near homoclinic orbits of the focussing nonlinear Schrödinger equation, preprint]. Thus, in this paper, the stationary equations of the CNLS are derived and the analytic structure of the trigonal spectral curve is examined, providing a basis for further study of the near homoclinic orbits of the CNLS system.

Bilinearization of Coupled Nonlinear Schrödinger Type Equations: Integrabilty and Solitons

Considering the coupled envelope equations in nonlinear couplers, the question of integrability is attempted. It is explicitly shown that Hirota’s bilinear method is one of the simple and alternative techniques to Painlevé analysis to obtain the integrability conditions of the coupled nonlinear Schrödinger (CNLS) type equations. We also show that the coupled Hirota equation introduced by Tasgal and Potasek is the next hierarchy of the inverse scattering solvable CNLS equation. The results are in agreement with the known results.

Superposition solutions of coupled nonlinear equations

Symmetrically coupled nonlinear Schrödinger (CNLS), coupled Korteweg-de Vries (CKdV), and coupled bound-continuum-bound (CBCB) equations are shown to have solitary-wave solutions that are generally expressible in terms of a superposition of two solitary waves of different forms. Control of wave propagation with different superpositions, and extending the superposition solutions to other nonlinearly coupled equations, are two possible useful applications.

Coupled NLS equations for counter propagating waves in systems with reflection symmetry

The amplitude of a wavetrain in a dispersive system generally evolves according to the cubic nonlinear Schrödinger (NLS) equation. In a reflection-symmetric system, the interaction of right and left travelling wavetrains with amplitudes ϵA +( ξ +, T 2) and ϵA -( ξ -, T 2) is described by two nonlinear Schrödinger equations with mean field coupling: ±2 i∂A ±/∂T 2+( d 2ω/ dk 2)×∂ 2A ±/∂ξ ±2=μ ±A ±+αA ±¦A ±¦ 2 . The independent variables are defined by ξ ±= ϵ( c g t∓ x ), T 2= ϵ 2 t and the coupling arises through μ ±≡β[(1/P ∓)∫ P∓ 0¦A ∓¦ 2 dξ ∓] . The analysis is applied to electromagnetic wave propagation in an optical fibre.

Isospectral Hamiltonian flows in finite and infinite dimensions

The approach to isospectral Hamiltonian flow introduced in part I is further developed to include integration of flows with singular spectral curves. The flow on finite dimensional Ad*-invariant Poisson submanifolds of the dual\((\widetilde{gl}(r)^ + )*\) of the positive part of the loop algebra\(\widetilde{gl}(r)\) is obtained through a generalization of the standard method of linearization on the Jacobi variety of the invariant spectral curveS. These curves are embedded in the total space of a line bundleT→P1(C), allowing an explicit analysis of singularities arising from the structure of the image of a moment map\(\tilde J_r :M_{N,r} \times M_{N,r} \to (\widetilde{gl}(r)^ + )*\) from the space of rank-r deformations of a fixedN×N matrixA. It is shown how the linear flow of line bundles\(E_t \to \tilde S\) over a suitably desingularized curve\(\tilde S\) may be used to determine both the flow of matricial polynomialsL(λ) and the Hamiltonian flow in the spaceMN,r×MN,r in terms of θ-functions. The resulting flows are proved to be completely integrable. The reductions to subalgebras developed in part I are shown to correspond to invariance of the spectral curves and line bundles\(E_t \to \tilde S\) under certain linear or anti-linear involutions. The integration of two examples from part I is given to illustrate the method: the Rosochatius system, and the CNLS (coupled non-linear Schrodinger) equation.

Similarity Solutions of the Two-Dimensional Coupled Nonlinear Schrodinger Equation

It is shown that the two-dimensional coupled nonlinear Schrodinger equation is reduced first to the coupled Klein-Gordon, one-dimensional nonlinear Schrodinger equation and others, and secondly to the second Painleve equation and others by similarity transformations. The similar-type soliton solutions of the equation are obtained.