An improved method for Darboux transformation for the coupled inhomogeneous nonlinear Schrödinger system from plasma physics and nonlinear optics

The coupled inhomogeneous nonlinear Schrödinger-type system which can be used to control soliton propagation and interaction in certain plasmas and optical fibers is investigated. An improved method for Darboux transformation (DT) is presented in more general forms by constructing an improved Γ$\Gamma$-Riccati-type Bäcklund transformation (Γ$\Gamma$-R BT). With the N$N$th-iterated Γ$\Gamma$-R BT or the N$N$th-iterated DT, which is a compact representation for the N$N$-soliton-like solutions and can generate a series of analytic solutions from a pair of the seed solutions through algebraic manipulations, the analytic one-/two-soliton-like solutions are provided. With the choice of parameters for the soliton solutions, the dynamical characteristics of the influences of the inhomogeneous parameters on the propagation of the soliton pulses are discussed graphically.     

Nondegenerate soliton solutions in certain coupled nonlinear Schrödinger systems

In this paper, we report a more general class of nondegenerate soliton solutions, associated with two distinct wave numbers in different modes, for a certain class of physically important integrable two component nonlinear Schrödinger type equations through bilinearization procedure. In particular, we consider coupled nonlinear Schrödinger (CNLS) equations (both focusing as well as mixed type nonlinearities), coherently coupled nonlinear Schrödinger (CCNLS) equations and long-wave-short-wave resonance interaction (LSRI) system. We point out that the obtained general form of soliton solutions exhibit novel profile structures than the previously known degenerate soliton solutions corresponding to identical wave numbers in both the modes. We show that such degenerate soliton solutions can be recovered from the newly derived nondegenerate soliton solutions as limiting cases.     

Effect of phase shift in shape changing collision of solitons in coupled nonlinear Schrödinger equations

Soliton interactions in systems modelled by coupled nonlinear Schr?dinger (CNLS) equations and encountered in phenomena such as wave propagation in optical fibers and photorefractive media possess unusual features: shape changing intensity redistributions, amplitude dependent phase shifts and relative separation distances. We demonstrate these properties in the case of integrable 2-CNLS equations. As a simple example, we consider the stationary two-soliton solution which is equivalent to the so-called partially coherent soliton (PCS) solution discussed much in the recent literature. Received 1st October 2001 / Received in final form 4 February 2002 Published online 2 October 2002 RID="a" ID="a"e-mail: lakshman@bdu.ernet.in     

New exact solution structures and nonlinear dispersion in the coupled nonlinear wave systems

In this Letter, to further understand the role of nonlinear dispersion in coupled nonlinear wave systems in both real and complex fields, we study the coupled Klein–Gordon equations with nonlinear dispersion in real field (called CKG(m,n,k)$\mathrm{CKG}(m,n,k)$ equation) and (2+1)$(2+1)$-dimensional generalization of coupled nonlinear Schrödinger equation with nonlinear dispersion in complex field (called GCNLS(m,n,k)$\mathrm{GCNLS}(m,n,k)$ equation) via some transformations. As a consequence, some types of solutions are obtained, which contain compactons, solitary pattern solutions, envelope compacton solutions, envelope solitary pattern solutions, solitary wave solutions and rational solutions.     

Curve flows in Lagrange–Finsler geometry,bi-Hamiltonian structures and solitons

Methods in Riemann–Finsler geometry are applied to investigate bi-Hamiltonian structures and related mKdV hierarchies of soliton equations derived geometrically from regular Lagrangians and flows of non-stretching curves in tangent bundles. The total space geometry and nonholonomic flows of curves are defined by Lagrangian semisprays inducing canonical nonlinear connections (N$N$-connections), Sasaki type metrics and linear connections. The simplest examples of such geometries are given by tangent bundles on Riemannian symmetric spaces G/SO(n)$G/SO\left(n\right)$ provided with an N$N$-connection structure and an adapted metric, for which we elaborate a complete classification, and by generalized Lagrange spaces with constant Hessian. In this approach, bi-Hamiltonian structures are derived for geometric mechanical models and (pseudo) Riemannian metrics in gravity. The results yield horizontal/vertical pairs of vector sine-Gordon equations and vector mKdV equations, with the corresponding geometric curve flows in the hierarchies described in an explicit form by nonholonomic wave maps and mKdV analogs of nonholonomic Schrödinger maps on a tangent bundle.     

拉曼增益对双折射光纤中孤子传输特性的影响

本文采用考虑拉曼增益的耦合非线性薛定谔方程,利用分步傅里叶方法求解并仿真模拟了光孤子脉冲在不同性质的双折射光纤中传输时的演化过程.结果表明,拉曼增益可以有效抑制非线性耦合导致的孤子漂移,同时会导致光孤子脉冲峰值在传输时不断增大,产生拉曼放大效应.拉曼增益也可以有效抑制双折射光纤中传输的相邻光孤子之间的相互作用.     

Particle-like platonic solutions in scalar gravity

We construct globally regular gravitating solutions, which possess only discrete symmetries. These solutions of Yang–Mills-dilaton theory may be viewed as exact (numerical) solutions of scalar gravity, by considering the dilaton as a kind of scalar graviton, or as approximate solutions of Einstein–Yang–Mills theory. We focus on platonic solutions with cubic symmetry, related to a rational map of degree N=4$N=4$. We present the first two solutions of the cubic N=4$N=4$ sequence, and expect this sequence to converge to an extremal Reissner–Nordström solution with magnetic charge P=4$P=4$.     

Generation of stable bright pulse and the dark soliton interaction in nonlinear left-handed materials

In homogeneous and isotropic nonlinear left-handed materials (LHMs), using the split step Fourier transform method, we demonstrate that two dark electromagnetic solitons which are the solutions of coupled nonlinear Schrödinger equations (CNLS) move along a line toward each other, when the distance between them down to a certain value, the interaction of the two dark solitons can produce a stable bright electromagnetic pulse that behaves like a bright soliton. Although our mathematical analysis shows that the new generated stable bright pulse is not an exact solution of the CNLS, it can propagate steadily in the nonlinear LHMs. And this unusual phenomenon can also be observed in our numerical simulation results by another method.     

Non-Abelian solutions in a Melvin magnetic universe

We show the existence of D=4$D=4$ non-Abelian solutions approaching asymptotically a dilatonic Melvin spacetime background. An exact solution generalizing the Chamseddine–Volkov soliton for a nonzero external U(1) magnetic field is also reported.     

A family of evolution equations with nonlinear diffusion,Verhulst growth,and global regulation: Exact time-dependent solutions

A family of evolution equations describing a power-law nonlinear diffusion process coupled with a local Verhulst-like growth dynamics, and incorporating a global regulation mechanism, is considered. These equations admit an interpretation in terms of population dynamics, and are related to the so-called conserved Fisher equation. Exact time-dependent solutions exhibiting a maximum nonextensive q$q$-entropy shape are obtained.     

Inverse scattering approach to coupled higher-order nonlinear Schrödinger equation and N-soliton solutions

A generalized inverse scattering method has been developed and applied to the linear problem associated with the coupled higher-order nonlinear Schrödinger equation to obtain it's N-soliton solution. An infinite number of conserved quantities have been obtained by solving a set of coupled Riccati equations. It has been shown that the coupled system admits two different class of solutions, characterized by the number of local maxima of amplitude of the soliton.     

Particle creation in asymptotically Minkowskian spacetimes

Exact solutions of Klein–Gordon and Dirac equations are obtained for two classes of Robertson–Walker (RW) spacetimes with asymptotically Minkowskian regions. One class is Minkowskian in the remote past and future. In this class in$in$ and out$out$ vacua are well defined, because the scale factor reduces to a constant at the asymptotic regions. Another class is asymptotically flat only in the far past. Using the obtained exact solutions we calculate the density of scalar and Dirac particles created through the Bogolubov transformations technique. For Dirac field it is shown that the rates of creation of particles and antiparticles are equal.     

Breather interactions and higher-order nonautonomous rogue waves for the inhomogeneous nonlinear Schrödinger Maxwell–Bloch equations

Under investigation in this paper are the inhomogeneous nonlinear Schrödinger Maxwell–Bloch (INLS-MB) equations which model the propagation of optical waves in an inhomogeneous nonlinear light guide doped with two-level resonant atoms. Higher-order nonautonomous breather as well as rogue wave solutions in terms of the determinants for the INLS-MB equations are presented via the n$n$-fold variable-coefficient modified Darboux transformation. The interactions among two nonautonomous breathers are graphically discussed, including the fundamental breather, bound breather, two-breather compression and two-breather evolution, etc. Moreover, several patterns of the higher-order rogue waves are also exhibited, such as the square rogue wave, two- and three-order periodic rogue waves, periodic fission and fusion, two-order stationary rogue waves, and recurrence of the two-order rogue waves. The character of the trajectory of the two-order periodic rogue wave is analyzed. Additionally, a novel type of interaction, namely, the collision between the breather and long-lived rogue waves, is found to be elastic. Our results could be useful for controlling the nonautonomous optical breathers and rogue waves in the inhomogeneous erbium doped fiber.     

Interaction of two dark solitons in nonlinear left-handed materials

A system of coupled nonlinear Schrödinger equations (CNLS) has dark soliton solutions. In nonlinear left-handed materials, the interaction of two of these dark solitons with the same parameter κ can generate new electromagnetic envelops that are quite different from a single soliton, even including a ‘bright’ envelop. But the new-generated envelops can also propagate steadily in the materials, we call them quasi-soliton. Numerical calculations demonstrate that varying either the parameter κ or the distance between the two solitons will change the envelop shape of the quasi-soliton remarkably. And an analysis of the phenomena is presented.     

Supercurrent coupling in the Faddeev–Skyrme model

Motivated by the sigma model limit of multicomponent Ginzburg–Landau theory, a version of the Faddeev–Skyrme model is considered in which the scalar field is coupled dynamically to a one-form field called the supercurrent. This coupled model is investigated in the general setting where physical space M$M$ is an oriented Riemannian manifold and the target space N$N$ is a Kähler manifold, and its properties are compared with the usual, uncoupled Faddeev–Skyrme model. In the case N=S²$N=S^2$, it is shown that supercurrent coupling destroys the familiar topological lower energy bound of Vakulenko and Kapitanski when M=R³$M={\mathbb{R}}^3$, and the less familiar linear bound when M$M$ is a compact 3-manifold. Nonetheless, local energy minimizers may still exist. The first variation formula is derived and used to construct three families of static solutions of the model, all on compact domains. In particular, a coupled version of the unit charge hopfion on a three-sphere of arbitrary radius is found. The second variation formula is derived, and used to analyze the stability of some of these solutions. In particular, it is shown that, in contrast to the uncoupled model, the coupled unit hopfion on the three-sphere of radius R$R$ is unstable   for all R$R$. This gives an explicit, exact example of supercurrent coupling destabilizing a stable solution of the usual Faddeev–Skyrme model, and casts doubt on the conjecture of Babaev, Faddeev and Niemi that knot solitons should exist in the low energy regime of two-component superconductors.     

The generalized non-linear Schrödinger model on the interval

The generalized (1+1)-D$(1+1)\text{-}\mathrm{D}$ non-linear Schrödinger (NLS) theory with particular integrable boundary conditions is considered. More precisely, two distinct types of boundary conditions, known as soliton preserving (SP) and soliton non-preserving (SNP), are implemented into the classical glN$g{l}_N$ NLS model. Based on this choice of boundaries the relevant conserved quantities are computed and the corresponding equations of motion are derived. A suitable quantum lattice version of the boundary generalized NLS model is also investigated. The first non-trivial local integral of motion is explicitly computed, and the spectrum and Bethe ansatz equations are derived for the soliton non-preserving boundary conditions.     

Hidden symmetry in the presence of fluxes

We derive the most general first-order symmetry operator for the Dirac equation coupled to arbitrary fluxes. Such an operator is given in terms of an inhomogeneous form ω   which is a solution to a coupled system of first-order partial differential equations which we call the generalized conformal Killing–Yano system. Except trivial fluxes, solutions of this system are subject to additional constraints. We discuss various special cases of physical interest. In particular, we demonstrate that in the case of a Dirac operator coupled to the skew symmetric torsion and U(1)$U(1)$ field, the system of generalized conformal Killing–Yano equations decouples into the homogeneous conformal Killing–Yano equations with torsion introduced in D. Kubiznak et al. (2009) [8] and the symmetry operator is essentially the one derived in T. Houri et al. (2010) [9]. We also discuss the Dirac field coupled to a scalar potential and in the presence of 5-form and 7-form fluxes.     

Some properties of the Alday–Maldacena minimum

The Alday–Maldacena solution, relevant to the n=4$n=4$ gluon amplitude in N=4$N=4$ SYM at strong coupling, was recently identified as a minimum of the regularized action in the moduli space of solutions of the AdS₅${\mathit{AdS}}_5$σ  -model equations of motion. Analogous solutions of the Nambu–Goto equations for the n=4$n=4$ case are presented and shown to form (modulo the reparametrization group) an equally large but different moduli space, with the Alday–Maldacena solution at the intersection of the σ  -model and Nambu–Goto moduli spaces. We comment upon the possible form of the regularized action for n=5$n=5$. A function of moduli parameters za$z_a$ is written, whose minimum reproduces the BDDK one-loop five-gluon amplitude. This function may thus be considered as some kind of Legendre transform of the BDDK formula and has its own value independently of the Alday–Maldacena approach.     

Classical space–times from the S-matrix

We show that classical space–times can be derived directly from the S-matrix for a theory of massive particles coupled to a massless spin two particle. As an explicit example we derive the Schwarzchild space–time as a series in _GN$G_N$. At no point of the derivation is any use made of the Einstein–Hilbert action or the Einstein equations. The intermediate steps involve only on-shell S-matrix elements which are generated via BCFW recursion relations and unitarity sewing techniques. The notion of a space–time metric is only introduced at the end of the calculation where it is extracted by matching the potential determined by the S-matrix to the geodesic motion of a test particle. Other static space–times such as Kerr follow in a similar manner. Furthermore, given that the procedure is action independent and depends only upon the choice of the representation of the little group, solutions to Yang–Mills (YM) theory can be generated in the same fashion. Moreover, the squaring relation between the YM and gravity three point functions shows that the seeds that generate solutions in the two theories are algebraically related. From a technical standpoint our methodology can also be utilized to calculate quantities relevant for the binary inspiral problem more efficiently then the more traditional Feynman diagram approach.