Derivative Nonlinear Schrodinger Equation Research Articles

Darboux and generalized Darboux transformations for the fractional integrable derivative nonlinear Schrödinger equation

Analytical methods provide crucial mathematical insights into the stable solitary waves hidden in nonlinear phenomena. The nonlinear Schrödinger (NLS) equation is one of the most important typical integrable soliton models. From a mathematical perspective, the essence of the celebrated derivative NLS (DNLS) equation’s difference from the classical NLS equation lies in its cubic potential being differentiated once by the spatial variable and multiplied by the imaginary unit, which leads to the former having some characteristics that the latter cannot have. This paper extends the DNLS equation to the fractional integrable case with conformable derivative operators, and uses Darboux transformations (DTs) and generalized DT (GDT) to solve it exactly. Specifically, Lax pairs generating the fractional DNLS equation are first given. Based on the given Lax pairs, then the n-fold DTs and GDT for the fractional DNLS equation are derived. Some special exact solutions of the fractional DNLS equation are further obtained by employing the derived n-fold DTs and GDT. Finally, several novel space–time structures and dynamical evolutions of the obtained exact solutions are analyzed. This paper reveals through the DT and GDT methods that the double power-law fractional orders in the exact solutions of the fractional DNLS equation can be used to dominate the variable velocity propagation and anomalous diffusion in fractional dimensional media at different geometric scales.

Dynamical stricture of optical soliton solutions and variational principle of nonlinear Schrödinger equation with Kerr law nonlinearity

The aim of this paper is to discover some new exact solutions for two kinds of nonlinear Schrödinger equation (NLSE), including the (2+1)-dimensional NLSE and the derivative NLSE, through the use of the variational principle method and amplitude ansatz method. We will find the solutions to these equations by selecting a trial function with a single nontrivial variational parameter, and it is continuous at all intervals in three cases from a region of a rectangular box, then using this trial function to obtain the functional integral and the Lagrangian of the system without any loss. After that, we approximate this trial function by quadratic polynomials with two free parameters rather than a piecewise linear ansatz function. Similarly, this trial function will be approximated by the tanh function. Additionally, we accept several new types of soliton solutions, including bright solitons, dark solitons, bright–dark solitary wave solutions, rational dark–bright solutions, and periodic solitary wave solutions. Also, conditions for the stability of the solutions will be proposed. These explanations are of great importance in the fields of applied science and engineering. The results will be shown through different types of graphs, including 2D, 3D, and contour plots.

Integrable reductions of the multi-component Kaup–Newell equations

We investigate the local and nonlocal integrable reductions of the multi-component Kaup–Newell equations and derive a range of novel integrable derivative nonlinear Schrödinger (NLS) equations. Firstly, we demonstrate that the multi-component Kaup–Newell equations can undergo one local reduction and three nonlocal reductions, specifically space-reversal, time-reversal, and space–time-reversal reductions. These reductions result in four multi-component derivative NLS equations. Secondly, focusing on the four-component Kaup–Newell equations, we introduce six joint reductions, which are combinations of the aforementioned local and nonlocal reductions. Through these joint reductions between the local reduction and nonlocal reductions, we discover three scalar nonlocal derivative NLS equations, including a T-reversal derivative NLS equation. Furthermore, through the joint reductions among the nonlocal reductions, we identify three scalar nonlocal derivative NLS equations with four variables, namely (x,t), (-x,t), (x,−t), and (−x,−t). All these reduced systems are integrable systems, meaning they possess infinitely many conserved integrals.

A robust way to justify the derivative NLS approximation

The derivative nonlinear Schrödinger (DNLS) equation can be derived as an amplitude equation via multiple scaling perturbation analysis for the description of the slowly varying envelope of an underlying oscillating and traveling wave packet in dispersive wave systems. It appears in the degenerated situation when the cubic coefficient of the similarly derived NLS equation vanishes. It is the purpose of this paper to prove that the DNLS approximation makes correct predictions about the dynamics of the original system under rather weak assumptions on the original dispersive wave system if we assume that the initial conditions of the DNLS equation are analytic in a strip of the complex plane. The method is presented for a Klein–Gordon model with a cubic nonlinearity.

Optical soliton solution via complete discrimination system approach along with bifurcation and sensitivity analyses for the Gerjikov-Ivanov equation

The aim of this research intends to investigate the complex wave patterns and dynamic behavior of the Gerjikov-Ivanov equation (GIE), commonly known as the derivative nonlinear Schrödinger equation (DNLSE). The GIE model is the one that examines dynamics of optical soliton propagation for transmission technologies, including transcontinental and transoceanic lengths of optical fibers, data transfer, and the telecommunications sector. The GIE equation can be regarded as an extension of the NLSE when certain higher-order nonlinear effects are taken into account. We study various optical soliton solutions, including Jacobian elliptic, hyperbolic, and trigonometric functions, are identified using the polynomial method’s complete discrimination system (CDSPM). In addition, Jacobian elliptic functions can be transformed into solitary wave solutions. We also discuss bifurcation points, sensitivity analysis and critical solution conditions, offering a thorough grasp of the system’s dynamic features. The CDSPM plays an important role in studying equilibrium points, phase diagrams, bifurcation behavior, and performing quantitative and qualitative evaluations.

Riemann–Hilbert approach to the focusing and defocusing nonlocal derivative nonlinear Schrödinger equation with step-like initial data

In this paper, we consider the Cauchy problem for the integrable nonlocal derivative nonlinear Schrödinger (DNLS) equation with step-like initial data. The main aim is to obtain a solution for the prescribed initial conditions. To do this, we adapt the Riemann–Hilbert (RH) approach to discuss of the step-like initial value problem associated with the nonlocal DNLS equation. The main idea is the representation of the solution of the Cauchy problem in terms of the solution of an associated 2 × 2 matrix RH problem and obtain the solution of the Cauchy problem of the nonlocal DNLS equation.

Study of a nonlinear Schrodinger equation with truncated M proportional derivative

In this work, we introduce a novel truncated M proportional (T-MP) derivative and consider a T-MP perturbed derivative nonlinear Schrodinger equation (PDNLSE). The PDNLSE is a nonlinear model which arises in nano optical fibers (photonic nanowires). While the DNLSE exhibits self-steepening (SS), the PDNLSE exhibits self-phase modulation (SPM) and Raman scattering (RS) effects, Here, the exact solutions of the PDENLSE are derived, here, by implementing the unified method. These solutions are displayed in graphs. It is found that a sufficient condition for a hyper-chaotic solution to hold is that an elliptic function solution exists. Non elliptic solutions may be, also, hyper-chaotic. It is worth mentioning that hyper chaotic may occur in economic and financial mathematics. Nonchaotic solutions exhibit many geometric structures, chirped solitons, and rhombus-shaped and M-shaped solitons. It is found that modulation instability triggers when the coefficient of the third-order dispersion exceeds a critical value. Further, the global bifurcation is investigated via phase portrait by constructing the Hamiltonian function.

The long-time asymptotics of the derivative nonlinear Schr[formula omitted]dinger equation with step-like initial value

Consideration in this present paper is the long-time asymptotics of solutions to the derivative nonlinear Schrödinger (DNLS) equation with the step-like initial value q(x,0)=q0(x)=A1eiϕe2iBx,x<0,A2e−2iBx,x>0,by Deift–Zhou method. The step-like initial problem is described by a matrix Riemann–Hilbert problem. A crucial ingredient used in this paper is to introduce the g-function mechanism for solving the problem of the entries of the jump matrix growing exponentially as t→∞. It is shown that the leading order term of the long-time asymptotics solution of the DNLS equation is expressed by the Theta function Θ about the Riemann-surface of genus 3 and the subleading order term expressed by parabolic cylinder and Airy functions.

Riemann–Hilbert problem and [formula omitted]-soliton solutions for the [formula omitted]-component derivative nonlinear Schrödinger equations

The n-component derivative nonlinear Schrödinger (DNLS) equations are discussed by the Riemann–Hilbert approach. By developing an improved Riemann–Hilbert problem, explicit soliton solutions of the n-component DNLS equations are obtained. Different from soliton solutions in many literatures, our expression does not include unfriendly integrals. An invariant and its general formula of the single-soliton solution for any n-component DNLS equations are derived. For the 2-component DNLS equation, we obtain one type of interesting solution — double hump soliton. The results play important roles in the vector solitons in many nonlinear physical systems, such as Bose–Einstein condensates, nonlinear optical fibers, etc.

Inverse scattering for the derivative nonlinear Schrödinger equation with nonzero boundary conditions and triple poles

The inverse scattering transform approach is used to investigate the derivative nonlinear Schrödinger equation (DNLSE) with nonzero boundary conditions (NZBCs) at infinity and triple zeros of analytical scattering coefficients. Based on the analytical, symmetric and asymptotic properties of eigenfunctions, a matrix Riemann–Hilbert problem (RHP) associated with DNLSE with NZBCs is constructed. Then, the reconstruction formula for the potential is found by solving the RHP. In particular, the reflectionless potential with triple poles for the NZBCs is carried out explicitly by means of determinants.

Complex excitations for the derivative nonlinear Schrödinger equation

The Darboux transformation (DT) formulae for the derivative nonlinear Schrödinger (DNLS) equation are expressed in concise forms, from which the multi-solitons, n-periodic solutions, higher-order hybrid-pattern solitons and some mixed solutions are obtained. These complex excitations can be constructed thanks to more general semi-degenerate DTs. Even the nondegenerate N-fold DT with a zero seed can generate complicated n-periodic solutions. It is proved that the solution q[N] at the origin depends only on the summation of the spectral parameters. We find the maximum amplitudes of several classes of the wave solutions are determined by the summation. Many interesting phenomena are discovered from these new solutions. For instance, the interactions between n-periodic waves produce peaks with different amplitudes and sizes. A soliton on a single-periodic wave background shares a similar feature as a breather due to the interference of the periodic background. In addition, the results are extended to the reverse-space-time DNLS equation.

Instability of degenerate solitons for nonlinear Schrödinger equations with derivative

We consider the following nonlinear Schrödinger equation with derivative: (1)iut=−uxx−i|u|2ux−b|u|4u,(t,x)∈R×R,b∈R.If b=0, this equation is a gauge equivalent form of the well-known derivative nonlinear Schrödinger (DNLS) equation. The soliton profile of DNLS satisfies a certain double power elliptic equation with cubic–quintic nonlinearities. The quintic nonlinearity in (1) only affects the coefficient in front of the quintic term in the elliptic equation, so in this sense the additional nonlinearity is natural as a perturbation preserving soliton profiles of DNLS. When b≥0, Eq. (1) has degenerate solitons whose momentum and energy are zero, and if b=0, they are algebraic solitons. Inspired from the works (Martel and Merle, 2001; Farah et al., 2019) on instability theory of the L2-critical generalized KdV equation, we study the instability of degenerate solitons of (1) in a qualitative way, and when b>0, we obtain a large set of initial data yielding the instability. The arguments except one step in our proof work for the case b=0 in exactly the same way, which is a small step towards understanding the dynamics around algebraic solitons of the DNLS equation.

Construction of multi-solitons and multi kink-solitons of derivative nonlinear Schrödinger equations

We look for solutions to derivative nonlinear Schrodinger equations built upon solitons. We prove the existence of multi-solitons i.e. solutions behaving at large time as the sum of finite solitons. We also show that one can attach a kink at the begin of the sum of solitons i.e. multi kink-solitons. Our proofs proceed by fixed point arguments around the desired profile, using Strichartz estimates.

Triple-pole soliton solutions of the derivative nonlinear Schrödinger equation via inverse scattering transform

We construct the triple-pole soliton solutions to the classical DNLS (derivative nonlinear Schrödinger) equation with zero boundary conditions at infinity through the inverse scattering transform method. Under the condition that the scattering coefficient has N-triple zeros, the detailed analysis of the discrete spectrum in direct problem is presented. The inverse problem is formulated and solved by means of the matrix Riemann–Hilbert problem with triple poles. As a consequence, we obtain the general solution of the DNLS equation. Moreover, the explicit N-triple-poles soliton formula for the reflectionless potential is derived.

Nonlinear Alfvén wave dynamics in positive ion -negative ion plasmas

The dynamical behavior of weakly nonlinear Alfvén waves is investigated in positive ion—negative ion (PINI) magnetized plasmas. The dynamics of the nonlinear wave is shown to be governed by a higher-order dispersion modified derivative nonlinear Schrödinger equation (DNLSE). The dispersive modification arises due to the inertial effect of both the ions. Analytical and numerical (for the experimental parameters) solutions predict the existence of solitary wave structures in positive ion—negative ion plasmas.

Alfvén solitons and generalized Darboux transformation for a variable-coefficient derivative nonlinear Schrödinger equation in an inhomogeneous plasma

Abstract Plasmas are believed to be possibly the most abundant form of visible matter in the Universe. Investigation in this paper is given to a variable-coefficient derivative nonlinear Schrodinger equation describing the propagation of the nonlinear Alfven waves in an inhomogeneous plasma. Based on the existing Lax pair, with respect to the left polarized Alfven wave, the N -fold generalized Darboux transformation, rational one/two Alfven soliton solutions and mixed two Alfven soliton solutions are derived, where N = 1 , 2 , 3 … . For those Alfven solitons, we find that (1) the solitonic width cannot be affected by the dispersion coefficient h ( τ ) and the loss/gain coefficient f ( τ ) , where τ is the stretched space variable; the solitonic amplitude grows (or decays) in the exponential rate exp [ − ∫ f ( τ ) d τ ] ; the solitonic amplitude cannot be affected by h ( τ ) ; the solitonic velocity and trajectory are related to h ( τ ) , while they cannot be affected by f ( τ ) ; (2) the rational two Alfven solitons experience no phase shift after the interaction; (3) the modulus square of mixed two Alfven solitons can be decomposed into two single Alfven solitons with the ∫ h ( τ ) d τ -dependent phase shifts; (4) the collapse of the rational one Alfven soliton is displayed.

Solving localized wave solutions of the derivative nonlinear Schrödinger equation using an improved PINN method

The solving of the derivative nonlinear Schrodinger equation (DNLS) has attracted considerable attention in theoretical analysis and physical applications. Based on the physics-informed neural network (PINN) which has been put forward to uncover dynamical behaviors of nonlinear partial different equation from spatiotemporal data directly, an improved PINN method with neuron-wise locally adaptive activation function is presented to derive localized wave solutions of the DNLS in complex space. In order to compare the performance of above two methods, we reveal the dynamical behaviors and error analysis for localized wave solutions which include one-rational soliton solution, genuine rational soliton solutions and rogue wave solution of the DNLS by employing two methods, also exhibit vivid diagrams and detailed analysis. The numerical results demonstrate the improved method has faster convergence and better simulation effect. On the bases of the improved method, the effects for different numbers of initial points sampled, residual collocation points sampled, network layers, neurons per hidden layer on the second order genuine rational soliton solution dynamics of the DNLS are considered, and the relevant analysis when the locally adaptive activation function chooses different initial values of scalable parameters are also exhibited in the simulation of the two-order rogue wave solution.

Rogue waves on the background of periodic standing waves in the derivative nonlinear Schrödinger equation.

The derivative nonlinear Schrödinger (DNLS) equation is the canonical model for the dynamics of nonlinear waves in plasma physics and optics. We study exact solutions describing rogue waves on the background of periodic standing waves in the DNLS equation. We show that the space-time localization of a rogue wave is only possible if the periodic standing wave is modulationally unstable. If the periodic standing wave is modulationally stable, the rogue wave solutions degenerate into algebraic solitons propagating along the background and interacting with the periodic standing waves. Maximal amplitudes of rogue waves are found analytically and confirmed numerically.

Chirped envelope solutions of short pulse propagation in highly nonlinear optical fiber

We consider the optical chirped envelope solutions of propagation of short pulse in highly nonlinear optical fibers, which is modeled by a kind of derivative nonlinear Schrodinger equation. We use the complete discrimination system for polynomial method to give a complete analysis of all chirped envelope solutions, and obtain a series of typical exact chirped patterns which include solitons, periodic solutions and singular solutions in terms of explicit and implicit functions.

Chirped envelope patterns of sup-pico-second pulse propagation through an optical fiber

We study the optical chirped envelope patterns of sub-pico-second pulse propagation through an optical fiber, which is modeled by the Kaup–Newell equation with the form of derivative nonlinear Schrodinger equation. The complete discrimination system for polynomial method is applied to give all chirped envelope solutions. A family of propagation patterns including the bright, dark, singular and elliptical type solutions are obtained.