General N‐Dark–Dark Solitons in the Coupled Nonlinear Schrödinger Equations

N‐dark–dark solitons in the integrable coupled NLS equations are derived by the KP‐hierarchy reduction method. These solitons exist when nonlinearities are all defocusing, or both focusing and defocusing nonlinearities are mixed. When these solitons collide with each other, energies in both components of the solitons completely transmit through. This behavior contrasts collisions of bright–bright solitons in similar systems, where polarization rotation and soliton reflection can take place. It is also shown that in the mixed‐nonlinearity case, two dark–dark solitons can form a stationary bound state.     

Multi-component vortex solutions in symmetric coupled nonlinear Schrödinger equations

A Hamiltonian system of incoherently coupled nonlinear Schrödinger (NLS) equations is considered in the context of physical experiments in photorefractive crystals and Bose-Einstein condensates. Due to the incoherent coupling, the Hamiltonian system has a group of various symmetries that include symmetries with respect to gauge transformations and polarization rotations. We show that the group of rotational symmetries generates a large family of vortex solutions that generalize scalar vortices, vortex pairs with either double or hidden charge, and coupled states between solitons and vortices. Novel families of vortices with different frequencies and vortices with different charges at the same component are constructed and their linearized stability problem is block-diagonalized for numerical analysis of unstable eigenvalues.     

Evans Function for Lax Operators with Algebraically Decaying Potentials

We study the instability of algebraic solitons for integrable nonlinear equations in one spatial dimension that include modified KdV, focusing NLS, derivative NLS, and massive Thirring equations. We develop the analysis of the Evans function that defines eigenvalues in the corresponding Lax operators with algebraically decaying potentials. The standard Evans function generically has singularities in the essential spectrum, which may include embedded eigenvalues with algebraically decaying eigenfunctions. We construct a renormalized Evans function and study bifurcations of embedded eigenvalues, when an algebraically decaying potential is perturbed by a generic potential with a faster decay at infinity. We show that the bifurcation problem for embedded eigenvalues can be reduced to cubic or quadratic equations, depending on whether the algebraic potential decays to zero or approaches a nonzero constant. Roots of the bifurcation equations define eigenvalues which correspond to nonlinear waves that are formed from unstable algebraic solitons. Our results provide precise information on the transformation of unstable algebraic solitons in the time-evolution problem associated with the integrable nonlinear equation. Algebraic solitons of the modified KdV equation are shown to transform to either travelling solitons or time-periodic breathers, depending on the sign of the perturbation. Algebraic solitons of the derivative NLS and massive Thirring equations are shown to transform to travelling and rotating solitons for either sign of the perturbation. Finally, algebraic homoclinic orbits of the focusing NLS equation are destroyed by the perturbation and evolve into time-periodic space-decaying solutions.     

Steady bifurcation and solitons in relativistic laser plasmas interaction

In this paper, steady bifurcation and solitons in relativistic laser plasmas interaction are investigated. At first, a new coupled equation for wake wave and the circularly polarized transversal electromagnetic wave is derived. It is a Hamiltonian system with two degrees of freedom. Then, a steady bifurcation analysis based on the coexistence of three different equilibrium states is given. Finally, a condition for predicting the existence of solitons is obtained in terms of the bifurcation control parameter and Hamiltonian function value. The soliton solutions are found numerically. It is shown that the solitons can exist in appropriate regime of vector potential frequency.     

Numerical method for finding 3D solitons of the nonlinear Schrödinger equation in the axially symmetric case

A system of two nonlinear Schrödinger equations is considered that governs the frequency doubling of femtosecond pulses propagating in an axially symmetric medium with quadratic and cubic nonlinearity. A numerical method is proposed to find soliton solutions of the problem, which is previously reformulated as an eigenvalue problem. The practically important special case of a single Schrödinger equation is discussed. Since three-dimensional solitons in the case of cubic nonlinearity are unstable with respect to small perturbations in their shape, a stabilization method is proposed based on weak modulations of the cubic nonlinearity coefficient and variations in the length of the focalizing layers. It should be emphasized that, according to the literature, stabilization was previously achieved by alternating layers with oppositely signed nonlinearities or by using nonlinear layers with strongly varying nonlinearities (of the same sign). In the case under study, it is shown that weak modulation leads to an increase in the length of the medium by more than 4 times without light wave collapse. To find the eigenfunctions and eigenvalues of the nonlinear problem, an efficient iterative process is constructed that produces three-dimensional solitons on large grids.     

Vector Solitons and Their Internal Oscilliations in Birefringent Nonlinear Optical Fibers

In this article, the vector solitons in birefringent nonlinear optical fibers are studied first. Special attention is given to the single-hump vector solitons due to evidences that only they are stable. Questions such as the existence, uniqueness, and total number of these solitons are addressed. It is found that the total number of them is continuously infinite and their polarizations can be arbitrary. Next, the internal oscillations of these vector solitons are investigated by the linearization method. Discrete eigenmodes of the linearized equations are identified. Such modes cause to the vector solitons a kind of permanent internal oscillations, which visually appear to be a combination of translational and width oscillations in the A and B pulses. The numerically observed radiation shelf at the tails of interacting pulses is also explained. Finally, the asymptotic states of the perturbed vector solitons are studied within both the linear and nonlinear theory. It is found that the state of internal oscillations of a vector soliton is always unstable. It invariably emits energy radiation and eventually evolves into a single-hump vector soliton state.     

Internal Oscillations and Radiation Damping of Vector Solitons

Internal modes of vector solitons and their radiation-induced damping are studied analytically and numerically in the framework of coupled nonlinear Schrödinger equations. Bifurcations of internal modes from the integrable systems are analyzed, and the region of their existence in the parameter space of vector solitons is determined. In addition, radiation-induced decay of internal oscillations is investigated. Both exponential and algebraic decay rates are identified.     

Extended Lagrangian approach for the defocusing nonlinear Schrödinger equation

We study the defocusing nonlinear Schrödinger (NLS) equation written in hydrodynamic form through the Madelung transform. From the mathematical point of view, the hydrodynamic form can be seen as the Euler–Lagrange equations for a Lagrangian submitted to a differential constraint corresponding to the mass conservation law. The dispersive nature of the NLS equation poses some major numerical challenges. The idea is to introduce a two‐parameter family of extended Lagrangians, depending on a greater number of variables, whose Euler–Lagrange equations are hyperbolic and accurately approximate NLS equation in a certain limit. The corresponding hyperbolic equations are studied and solved numerically using Godunov‐type methods. Comparison of exact and asymptotic solutions to the one‐dimensional cubic NLS equation (“gray” solitons and dispersive shocks) and the corresponding numerical solutions to the extended system was performed. A very good accuracy of such a hyperbolic approximation was observed.     

Coherent Structures in Weakly Birefringent Nonlinear Optical Fibers

This article studies two coupled nonlinear Schrodinger equations that govern the pulse propagation in weakly birefringent nonlinear optical fibers. The coherent structures for these equations, such as vector solitons and localized oscillating solutions, are studied analytically and numerically. Three types of localized oscillating structures are identified and their functional forms determined by perturbation methods. In some of these structures, infinite oscillating tails are present. The implications of these tails are also discussed.     

Effect of inhomogeneity in energy transfer through alpha helical proteins with interspine coupling

The dynamics of homogeneous and inhomogeneous alpha helical proteins with interspine coupling is under investigation in this paper by proposing a suitable model Hamiltonian. For specific choice of parameters, the dynamics of homogeneous alpha helical proteins is found to be governed by a set of completely integrable three coupled derivative nonlinear Schrödinger (NLS) equations (Chen–Lee–Liu equations). The effect of inhomogeneity is understood by performing a perturbation analysis on the resulting perturbed three coupled NLS equation. An equivalent set of integrable discrete three coupled derivative NLS equations is derived through an appropriate generalization of the Lax pair of the original Ablowitz–Ladik lattice and the nature of the energy transfer along the lattice is studied.     

Optical Soliton Bullets in (2+1)D Nonlinear Bragg Resonant Periodic Geometries

We consider light propagation in a Kerr-nonlinear 2D waveguide with a Bragg grating in the propagation direction and homogeneous in the transverse direction. Using Newton's iteration method we construct both stationary and travelling solitary wave solutions of the corresponding mathematical model, the 2D nonlinear coupled mode equations (2D CME). We call these solutions 2D gap solitons due to their similarity with the gap solitons of 1D CME (fiber grating). Long-time stable evolution preserving the solitary fashion is demonstrated numerically despite the fact that, as we show, for the 2D CME no local constrained minima of the Hamiltonian functional exist. Building on the 1D study of [ 1 ], we demonstrate trapping of slow enough 2D gap solitons at localized defects. We explain the mechanism of trapping as resonant transfer of energy from the soliton to one or more nonlinear defect modes. For a special class of defects, we construct a family of nonlinear defect modes by numerically following a bifurcation curve starting at analytically or numerically known linear defect modes. Compared to 1D the dynamics of trapping are harder to fully analyze and the existence of many defect modes for a given defect potential causes that slow solitons store a part of their energy for virtually all of the studied attractive defects.     

Exact soliton solutions of a D-dimensional nonlinear Schr?dinger equation with damping and diffusive terms

In the present study, we apply function transformation methods to the D-dimensional nonlinear Schr?dinger (NLS) equation with damping and diffusive terms. As special cases, this method applies to the sine-Gordon, sinh-Gordon, and other equations. Also, the results show that these equations depend on only one function that can be obtained analytically by solving an ordinary differential equation. Furthermore, certain exact solutions of these three equations are shown to lead to the exact soliton solutions of a D-dimensional NLS equation with damping and diffusive terms. Finally, our results imply that the planar solitons, N multiple solitons, propagational breathers, and quadric solitons are solutions to the sine-Gordon, sinh-Gordon, and D-dimensional NLS equations.     

Nonlinear matrix algebra and engineering applications. Part 1: Theory and linear form matrix

A matrix vector formalism is developed for systematizing the manipulation of sets of non-linear algebraic equations. In this formalism all manipulations are performed by multiplication with specially constructed transformation matrices. For many important classes of nonlinearities, algorithms based on this formalism are presented for rearranging a set of equations so that their solution may be obtained by numerically searching along a single variable. Theory developed proves that all solutions are obtained.     

Nonlinear matrix algebra and engineering applications Part 2: Polynomial form matrices

A matrix vector formalism is developed for systematizing the manipulation of sets of nonlinear algebraic equations. In this formalism all manipulations are performed by multiplication with specially constructed transformation matrices. For many important classes of nonlinearities, algorithms based on this formalism are presented for rearranging a set of equations so that their solution may be obtained by numerically searching along a single variable. Theory developed proves that all solutions are obtained.     

Exact soliton solutions of a D-dimensional nonlinear Schrödinger equation with damping and diffusive terms

In the present study, we apply function transformation methods to the D-dimensional nonlinear Schrödinger (NLS) equation with damping and diffusive terms. As special cases, this method applies to the sine-Gordon, sinh-Gordon, and other equations. Also, the results show that these equations depend on only one function that can be obtained analytically by solving an ordinary differential equation. Furthermore, certain exact solutions of these three equations are shown to lead to the exact soliton solutions of a D-dimensional NLS equation with damping and diffusive terms. Finally, our results imply that the planar solitons, N multiple solitons, propagational breathers, and quadric solitons are solutions to the sine-Gordon, sinh-Gordon, and D-dimensional NLS equations.     

Relaxation of solitons in nonlinear Schrödinger equations with potential

In this paper we study dynamics of solitons in the generalized nonlinear Schrödinger equation (NLS) with an external potential in all dimensions except for 2. For a certain class of nonlinearities such an equation has solutions which are periodic in time and exponentially decaying in space, centered near different critical points of the potential. We call those solutions which are centered near the minima of the potential and which minimize energy restricted to L²-unit sphere, trapped solitons or just solitons. In this paper we prove, under certain conditions on the potentials and initial conditions, that trapped solitons are asymptotically stable. Moreover, if an initial condition is close to a trapped soliton then the solution looks like a moving soliton relaxing to its equilibrium position. The dynamical law of motion of the soliton (i.e. effective equations of motion for the soliton's center and momentum) is close to Newton's equation but with a dissipative term due to radiation of the energy to infinity.     

Darboux transformation and soliton solutions for the coupled cubic-quintic nonlinear Schrödinger equations in nonlinear optics

In this paper, by virtue of the Darboux transformation (DT) and symbolic computation, the quintic generalization of the coupled cubic nonlinear Schrödinger equations from twin-core nonlinear optical fibers and waveguides are studied, which describe the effects of quintic nonlinearity on the ultrashort optical pulse propagation in non-Kerr media. Lax pair of the equations is obtained and the corresponding DT is constructed. Moreover, one-, two- and three-soliton solutions are presented in the forms of modulus. Features of solitons are graphically discussed: (1) head-on and overtaking elastic collisions of the two solitons; (2) periodic attraction and repulsion of the bounded states of two solitons; (3) energy-exchanging collisions of the three solitons.     

A direct algebraic method applied to obtain complex solutions of some nonlinear partial differential equations

By using some exact solutions of an auxiliary ordinary differential equation, a direct algebraic method is described to construct the exact complex solutions for nonlinear partial differential equations. The method is implemented for the NLS equation, a new Hamiltonian amplitude equation, the coupled Schrodinger–KdV equations and the Hirota–Maccari equations. New exact complex solutions are obtained.     

Asymptotic Analysis of Pulse Dynamics in Mode-Locked Lasers

Solitons of the power-energy saturation (PES) equation are studied using adiabatic perturbation theory. In the anomalous regime individual soliton pulses are found to be well approximated by solutions of the classical nonlinear Schrödinger (NLS) equation with the key parameters of the soliton changing slowly as they evolve. Evolution equations are found for the pulse amplitude(s), velocity(ies), position(s), and phase(s) using integral relations derived from the PES equation. The results from the integral relations are shown to agree with multi-scale perturbation theory. It is shown that the single soliton case exhibits mode-locking behavior for a wide range of parameters, while the higher states form effective bound states. Using the fact that there is weak overlap between tails of interacting solitons, evolution equations are derived for the relative amplitudes, velocities, positions, and phase differences. Comparisons of interacting soliton behavior between the PES equation and the classical NLS equation are also exhibited.     

Vibration reduction in a 2DOF twin-tail system to parametric excitations

The use of active feedback control strategy is a common way to stabilize and control dangerous vibrations in vibrating systems and structures, such as bridges, highways, buildings, space and aircrafts. These structures are distributed-parameter systems. Unfortunately, the existing vibrations control techniques, even for these simplified models, are fraught with numerical difficulties and engineering limitations. In this paper, a negative velocity feedback is added to the dynamical system of twin-tail aircraft, which is represented by two coupled second-order nonlinear differential equations having both quadratic and cubic nonlinearities. The system describes the vibration of an aircraft tail subjected to multi-parametric excitation forces. The method of multiple time scale perturbation is applied to solve the nonlinear differential equations and obtain approximate solutions up to the third order approximations. The stability of the system is investigated applying frequency response equations. The effects of the different parameters are studied numerically. Some different resonance cases are investigated. A comparison is made with the available published work.