Coupled Nonlinear Schrodinger Equations Research Articles

Structural stability of non-ground state traveling waves of coupled nonlinear Schrödinger equations

We consider the linear stability and structural stability of non-ground state traveling waves of a pair of coupled nonlinear Schrödinger equations (CNLS) which describe the evolution of co-propagating polarized pulses in the presence of birefringence. Viewing the CNLS equations as a Hamiltonian perturbation of the Manakov equations, we find parameter regimes in which there are two stable families of traveling waves. The usual variational methods for stability analysis of ground states do not apply. Instead we employ a Liapunov-Schmidt type reduction to detect eigenvalues bifurcating from the imaginary axis. We also demonstrate the instability of a family of vector solitons.

Envelope solitary waves on two-dimensional lattices with in-plane displacements

We investigate envelope solitary waves on square lattices with two degrees of freedom and nonlinear nearest and next-nearest neighbor interactions. We consider solitary waves which are localized in the direction of their motion and periodically modulated along the perpendicular direction. In the quasi-monochromatic approximation and low-amplitude limit a system of two coupled nonlinear Schrodinger equations (CNLS) is obtained for the envelopes of the longitudinal and transversal displacements. For the case of bright envelope solitary waves the solvability condition is discussed, also with respect to the modulation. The stability of two special solution classes (type-I and type-II) of the CNLS equations is tested by molecular dynamics simulations. The shape of type-I solitary waves does not change during propagation, whereas the width of type-II excitations oscillates in time.

Influence of dispersive waves on vector soliton propagation

AbstractA perturbation theory for the coupled nonlinear Schrödinger (NLS) equations is developed to investigate the effect of the background radiation on the soliton amplitude, frequency and polarization angle. By resorting to the theory of the inverse scattering transform, it is shown that the background radiation can be taken into account by evaluating the off diagonal terms of the scattering matrix associated with the coupled NLS equations. The theory is applied to the characterization of the soliton amplitude decrease observed in the propagation in the presence of loss and periodical amplification.

Soliton aspects of the inhomogeneous coupled nonlinear Schrödinger equation in optical fibre

The inhomogeneous coupled nonlinear Schrodinger (ICNLS) equations which describe the propagation of two fields in a nonuniform fibre medium are considered. The integrable forms of the ICNLS equation are identified from the Painleve singularity structure analysis. The variable transformations which connect the integrable forms of the ICNLS equation and the coupled nonlinear Schrodinger equations (CNLS) are presented. With the help of the linear eigenvalue problem and Backlund transformation, a single-soliton solution is generated explicitly.

Bright and dark soliton solutions to coupled nonlinear Schrodinger equations

A set of two coupled nonlinear Schrodinger equations (CNLS) describes the two mode propagation in an optical fibre. The Painleve singularity structure analysis singles out two integrable parametric choices, including the Manakov model. Using the results of Painleve analysis, we succeed in Hirota bilinearizing the CNLS equations in both the anomalous as well as normal dispersion regions for the integrable cases. Solving the Hirota bilinear equations, bright and dark N-soliton solutions are explicitly obtained.

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.

Painleve analysis and integrability of coupled non-linear Schrodinger equations

Considering a system of coupled non-linear Schrodinger (NLS) equations, the authors discuss the integrability properties through Painleve (P) analysis. For the two coupled NLS equations, they show that there exists a pair of parametric values possessing the P property, for which the associated Backlund transformation (BT) and the Hirota bilinearisation are constructed. These parametric choices are identical to those of Zakharov and Schulman (1982) who established the integrability in terms of 'motion invariants'. Finally, they extend the P analysis to the N coupled NLS system, and identify two parametric choices possessing the P property, which are natural generalisations of the two coupled NLS cases.

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.