Derivative Nonlinear Schrodinger Equation Research Articles

A newly revised inverse scattering transform for DNLS+ equation under nonvanishing boundary condition

A newly revised inverse scattering transform (IST) for the derivative nonlinear Schrodinger (DNLS+) equation with non-vanishing boundary condition (NVBC) and normal group velocity dispersion is proposed by introducing a suitable affine parameter in Zakharov-Shabat integral kern. The explicit breather-type one-soliton solution, which can reproduce one pure soliton at the degenerate case and one bright soliton solution at the limit of vanishing boundary, has been derived to verify the validity of the revised IST.

Quasi-periodic solutions for derivative nonlinear Schrödinger equation

In this paper, we discuss the existence of time quasi-periodic solutions for the derivative nonlinear Schrodinger equation $$\label{1.1}\mathbf{i} u_t+u_{xx}+\mathbf{i} f(x,u,\bar{u})u_x+g(x,u,\bar{u})=0 $$ subject to Dirichlet boundary conditions. Using an abstract infinite dimensional KAM theorem dealing with unbounded perturbation vector-field and Birkhoff normal form, we will prove that there exist a Cantorian branch of KAM tori and thus many time quasi-periodic solutions for the above equation.

A KAM Theorem for Hamiltonian Partial Differential Equations with Unbounded Perturbations

We establish an abstract infinite dimensional KAM theorem dealing with unbounded perturbation vector-field, which could be applied to a large class of Hamiltonian PDEs containing the derivative ∂ x in the perturbation. Especially, in this range of application lie a class of derivative nonlinear Schrodinger equations with Dirichlet boundary conditions and perturbed Benjamin-Ono equation with periodic boundary conditions, so KAM tori and thus quasi-periodic solutions are obtained for them.

Integrable twofold hierarchy of perturbed equations and application to optical soliton dynamics

We construct well-known integrable equations with their Lax pairs from the corresponding linear equations using our nonlinearization scheme. Using negative powers in the spectral flow to deform the time Lax operator, we find a class of perturbations that unlike the usual perturbations, which spoil the system integrability, exhibit a twofold integrable hierarchy, including those for the KdV, modified KdV, sine-Gordon, nonlinear Schrodinger (NLS), and derivative NLS equations. We discover hidden possibilities of using the perturbed hierarchy of the NLS equations to amplify and control optical solitons propagating through a fiber in a doped nonlinear resonant medium.

Solitonic interactions, Darboux transformation and double Wronskian solutions for a variable-coefficient derivative nonlinear Schrödinger equation in the inhomogeneous plasmas

By symbolic computation we study a variable-coefficient derivative nonlinear Schrodinger (vc-DNLS) equation describing nonlinear Alfven waves in inhomogeneous plasmas. Based on the Lax pair of the vc-DNLS equation, the N-fold Darboux transformation is constructed via a gauge transformation and the reduction technique. Multi-solitonic solutions in terms of the double Wronskian for the vc-DNLS equation are obtained. Two- and three-solitonic interactions are analyzed graphically, i.e., overtaking, head-on and parallel interactions. Plasma streaming and inhomogeneous magnetic field control the amplitudes and velocities of the solitonic waves, respectively. The nonuniform density affects the amplitudes of the solitonic waves. The effects of the spectral parameters on the dynamics of the two-solitonic waves are discussed. Our results might facilitate the analytic investigation on certain inhomogeneous systems in the Earth’s magnetosphere, solar winds, planetary bow shocks, dusty cometary tails and interplanetary shocks.

On the Cauchy problem for the nonlocal derivative nonlinear Schrödinger equation

We consider the Cauchy problem associated with the one-dimensional nonlocal derivative nonlinear Schrodinger equation, and establish local well-posedness for “small” initial data in the usual L-based Sobolev spaces H(R), s > 1/2. We also prove that our result is “almost sharp” in the sense that the flow-map data-solution fails to be C at the origin from H(R) to H(R) for any s < 1/2. Finally, thanks to the lack of energy conservation, we prove the nonexistence of solitary-wave solutions.

Breather and double-pole solutions of the derivative nonlinear Schrödinger equation from optical fibers

The derivative nonlinear Schrödinger (DNLS) equation, which governs the propagation of the femtosecond optical pulse in a monomodal optical fiber, is analytically studied in this Letter. Breather and double-pole solutions are derived from the two-soliton solution with the choice of parameters. It is found that the parameters in the DNLS equation cannot only control the phase and propagation direction of the breather and double pole, but also influence the interaction period of the breather. Elastic collisions between a breather and a dark/anti-dark soliton are studied by the qualitative analysis and graphical illustration. The stability of the breather and double-pole solutions is also analyzed.

The solution of the global relation for the derivative nonlinear Schrödinger equation on the half-line

We consider initial-boundary value problems for the derivative nonlinear Schrödinger (DNLS) equation on the half-line x > 0 . In a previous work, we showed that the solution q ( x , t ) can be expressed in terms of the solution of a Riemann–Hilbert problem with jump condition specified by the initial and boundary values of q ( x , t ) . However, for a well-posed problem, only part of the boundary values can be prescribed; the remaining boundary data cannot be independently specified, but are determined by the so-called global relation. In general, an effective solution of the problem therefore requires solving the global relation. Here, we present the solution of the global relation in terms of the solution of a system of nonlinear integral equations. This also provides a construction of the Dirichlet-to-Neumann map for the DNLS equation on the half-line.

Soliton-like solutions of a derivative nonlinear Schrödinger equation with variable coefficients in inhomogeneous optical fibers

Under investigation in this paper is a derivative nonlinear Schrodinger equation with variable coefficients, which governs the propagation of the subpicosecond soliton pulses in inhomogeneous optical fibers. Through the nonisospectral Kaup–Newell scheme, the Lax pair is constructed with some constraints on the variable coefficients. Under the integrable conditions, bright one- and multi-soliton-like solutions are derived via the Hirota method. By suitably choosing the dispersion coefficient function, several types of inhomogeneous solitons are obtained in, respectively: (1) exponentially decreasing dispersion profile, (2) linearly decreasing dispersion profile, (3) exponentially increasing dispersion profile, and (4) periodically fluctuating dispersion profile. The intensity of the inhomogeneous soliton can be controlled by means of modifying the loss/gain term. Asymptotic analysis of the two-soliton-like solution is performed, which shows that the changes of the widths, amplitudes, and energies before and after the collision are completely caused by the variable coefficients, but have nothing to do with the collision between two soliton-like envelopes. Through suitable choices of variable coefficients, figures are plotted to illustrate the collision behavior between two inhomogeneous solitons, which has some potential applications in the real optical communication systems.

A multi-soliton solution of the DNLS equation based on pure Marchenko formalism

By means of some algebraic techniques, especially the Binet-Cauchy formula, an explicit multi-soliton solution of the derivative nonlinear Schrodinger equation with vanishing boundary condition is attained based on a pure Marchenko formalism without needing the usual scattering data except for given N simple poles. The one- and two-soliton solutions are given as two special examples in illustration of the general formula of multi-soliton solution. Their effectiveness and equivalence to other approaches are also demonstrated. Meanwhile, the asymptotic behavior of the multi-soliton solution is discussed in detail. It is shown that the N-soliton solution can be viewed as a summation of N one-soliton solutions with a definite displacement and phase shift of each soliton in the whole process (from t → −∞ to t → +∞) of the elastic collisions.

Global Well-Posedness of the Derivative Nonlinear Schroedinger Equations on T

We prove the global well-posedness for the Cauchy problem of the derivative nonlinear Schrodinger equation in Hs(T) for s > 1/2 with small data in L2. We use the method of almost conserved energy or the I-method which was introduced by Colliander et al. and refine the bilinear estimate.

Soliton solution of the DNLS equation based on Hirota’s bilinear derivative transform

Based on the method of Hirota’s bilinear derivative transform, the derivative nonlinear Schrodinger equation with vanishing boundary condition has been directly solved. The one- and two-soliton solutions are given as two typical examples in the illustration of the general procedures and the concrete cut-off technique of the series-form solution, and the n-soliton solution is also attained by induction method. Our study shows their equivalence to the existing soliton solutions by a simple parameter transformation. The methodological importance of bilinear derivative transform in dealing with an integrable nonlinear equation has also been emphasized. The evolution of one and two-soliton solution with respect to time and space has been discussed in detail. The collision among the solitons has been manifested through an example of two-soliton case, revealing the elastic essence of the collision and the invariance of the soliton form and characteristics.

Evolution of Alfvénic wave envelopes in spin-1/2 quantum Hall-magnetohydrodynamic plasmas

The one-dimensional oblique propagation of large amplitude magnetohydrodynamic (MHD) waves in a high-β quantum Hall-MHD plasma is studied with electron spin-1/2 effects. The plasma β becomes high by the condition for the nonrelativistic fluid model to be valid and the condition for the collective effects to be important in quantum plasmas. Such a high-β value is a prerequisite for large perturbations of the perpendicular magnetic field comparable with the longitudinal magnetic field. It is shown that the nonlinear evolution of such waves is described by a derivative nonlinear Schrödinger (DNLS) equation. It is found that the DNLS equation does not depend on the higher order quantum coupling associated with the Bohm potential, rather the pressure such as spin force plays the crucial role. Such an evolution equation is shown to admit spin-modified localized envelope solitons whose width L is reduced by ε2/vB2 and the amplitude increases with increasing ε2/vB2 values, where ε is the temperature normalized Zeeman energy and vB2 is the electron thermal energy normalized by the Alfvén wave energy. Moreover, the MHD waves are found to be modulationally unstable for a wave number exceeding its critical value, which typically depends on ε2/vB2. The growth rate of the modulational instability is also investigated. Furthermore, the effect of dissipation due to plasma resistivity is shown to exhibit envelope shocklike structures instead of envelope solitons. The present nonlinear excitations can account for large scale structures in dense astrophysical plasma environments.

Topological study of multiple coexisting attractors in a nonlinear system

Multiple attractors in a new nonlinear dynamical system recently derived from the derivative nonlinear Schrodinger (DNLS) equation describing the Alfven turbulence in magnetized plasma are analyzed in the light of geometric characterization of orbit properties, which was originally introduced by Birman and Williams and later systematized by Gilmore, Mindlin, Letellier and others. We observe that the order in which periodic orbits appear is as predicted from their symbolic sequences by the universal sequence. The observed templates of increasing complexity are organized along a spiral (which is actually a two-dimensional homomorphic projection of the original attractor). Examples of explicit templates and their variation with respect to the parameter are exhibited.

Variational method for the derivative nonlinear Schrödinger equation with computational applications

The derivative nonlinear Schrödinger equation (DNLSE) arises as a physical model for ultra-short pulse propagation. In this paper, the existence of a Lagrangian and the invariant variational principle (i.e. in the sense of the inverse problem of calculus of variations through deriving the functional integral corresponding to a given coupled nonlinear partial differential equations) for two-coupled equations describing the nonlinear evolution of the Alfvén wave with magnetosonic waves at a much larger scale are given and the functional integral corresponding to those equations is derived. We found the solutions of DNLSE by choice of a trial function in a region of a rectangular box in two cases, and using this trial function, we find the functional integral and the Lagrangian of the system without loss. Solution of the general case for the two-box potential can be obtained on the basis of a different ansatz where we approximate the Jost function using polynomials of order n instead of the piecewise linear function. An example for the third order is given for illustrating the general case.

Exactly Solvable Model for Nonlinear Pulse Propagation in Optical Fibers

The nonlinear Schrödinger (NLS) equation is a fundamental model for the nonlinear propagation of light pulses in optical fibers. We consider an integrable generalization of the NLS equation, which was first derived by means of bi‐Hamiltonian methods in [1]. The purpose of the present paper is threefold: (a) We show how this generalized NLS equation arises as a model for nonlinear pulse propagation in monomode optical fibers when certain higher‐order nonlinear effects are taken into account; (b) We show that the equation is equivalent, up to a simple change of variables, to the first negative member of the integrable hierarchy associated with the derivative NLS equation; (c) We analyze traveling‐wave solutions.

Chiral solitons in a quantum potential

We study chiral solitons in a quantum potential using a dimensional reduction of the problem for (2+1)-dimensional anyons. We show that the integrable version of the model is described by a family of the resonant derivative nonlinear Schrodinger equations. For a quantum potential strength s > 1, we show that the chiral soliton interaction has a resonance. We investigate the semiclassical quantization procedure for solitons.

Madelung fluid description of the generalized derivative nonlinear Schrödinger equation: Special solutions and their stability

A correspondence between the families of generalized nonlinear Schrodinger (NLS) equations and generalized KdV equations was recently found using a Madelung fluid description. We similarly consider a special derivative NLS equation. We find a number of solitary waves and periodic solutions (expressed in terms of elliptic Jacobi functions) for a motion with a stationary profile current velocity. We study the stability of a bright solitary wave (ground state) by conjecturing that the Vakhitov-Kolokolov criterion is applicable.

The truncation model of the derivative nonlinear Schrödinger equation

The derivative nonlinear Schrödinger (DNLS) equation is explored using a truncation model with three resonant traveling waves. In the conservative case, the system derives from a time-independent Hamiltonian function with only one degree of freedom and the solutions can be written using elliptic functions. In spite of its low dimensional order, the truncation model preserves some features from the DNLS equation. In particular, the modulational instability criterion fits with the existence of two hyperbolic fixed points joined by a heteroclinic orbit that forces the exchange of energy between the three waves. On the other hand, numerical integrations of the DNLS equation show that the truncation model fails when wave energy is increased or left-hand polarized modulational unstable modes are present. When dissipative and growth terms are added the system exhibits a very complex dynamics including appearance of several attractors, period doubling bifurcations leading to chaos as well as other nonlinear phenomenon. In this case, the validity of the truncation model depends on the strength of the dissipation and the kind of attractor investigated.

Magnetic hole formation from the perspective of inverse scattering theory

[1] The dynamics of oblique, weakly dispersive nonlinear Alfven waves in the presence of weak resistive damping are investigated numerically through an extension of the derivative nonlinear Schrodinger (DNLS) equation. It is observed numerically that the nonlinear dynamics are organized around the dynamics and allowed interactions of the underlying DNLS soliton families. There are three types of oblique Alfven solitons: the compressive two-parameter soliton and one-parameter bright soliton along with the rarefactive one-parameter dark soliton. The damping of either of these compressive solitons is accompanied by the formation of one or more dark solitons. The implication of these processes is that any initial wave profile containing solitons in its Inverse Scattering Transformation representation, in the presence of weak resistive damping, will result in a leading train of dark solitons. These dark solitons have been identified with magnetic holes, and the results described above are discussed in the context of magnetic hole observations and theory.