Higher-order Nonlinear Schrodinger Equation Research Articles

Nonexistence of solutions of higher-order nonlinear non-Gauge Schrödinger equation

A nonexistence result is proved of the space higher-order nonlinear Schrodinger equation where m > 1, n > 1 and p > n. Our method of proof rests on a judicious choice of the test function in the weak formulation of the equation. Then, we obtain an upper bound of the life span of solutions. Furthermore, the necessary conditions for the existence of local or global solutions are provided.Next, we extend our results to the 2 × 2 – system.

Spectral and soliton structures of the Sasa–Satsuma higher-order nonlinear Schrödinger equation

In this paper, the Riemann–Hilbert problem of the Sasa–Satsuma higher-order nonlinear Schrodinger equation is investigated, from which spectral and soliton structures are discussed in detail. In addition, an algebra technique is developed to illustrate the soliton structures using Mathematica symbolic computations by choosing suitable parameters, including breather-bell soliton, double-hump soliton, and bell-to-breather soliton interactions. The results show that the spectral and soliton structures of the Sasa–Satsuma higher-order nonlinear Schrodinger equation are more complicated than many other nonlinear Schrodinger type equations.

Riemann–Hilbert approach and nonlinear dynamics of the coupled higher-order nonlinear Schrödinger equation in the birefringent or two-mode fiber

The multi-soliton solutions and breathers to the coupled higher-order nonlinear Schrodinger (CH-NLS) equation are derived in this work via the Riemann–Hilbert approach. Firstly, the spectral structure of the CH-NLS equation is investigated and then a matrix Riemann–Hilbert problem on the real axis is strictly formulated. Secondly, by solving the special Riemann–Hilbert problem with no reflection where a jump matrix is taken to be the identity matrix, the formula of N-soliton solutions can be computed. Thirdly, we prove that the higher-order linear and nonlinear term r has important impact on the velocity, phase, period and wavewidth of wave dynamics. Besides, the localized waves characteristics together with collision dynamic behaviors of these explicit soliton solutions and breathers are shown graphically and discussed in detail. Interestingly, three solitons display different dynamics which demonstrate amplitudes of the right-direction waves gradually become larger during the propagation process.

Dark-managed solitons in inhomogeneous cubic–quintic–septimal nonlinear media

We investigate the inhomogeneous higher-order nonlinear Schrodinger (INHLS) equation including cubic–quintic–septic (CQS) nonlinear terms and gain or loss with variable coefficients. The exact analytic solution that describes dark soliton-type pulse propagation is found for the model by employing the ansatz method. Unlike the traditional $$\text {tanh}$$ dark soliton in Kerr-type media, the functional form of this novel dark-managed soliton structure takes a $$\text {sech}^{2/3}$$ profile. Parametric conditions are presented in which these optical solitons exist. We also investigated the stability of these dark-managed solitons under some initial perturbations by employing the numerical simulation methods. Finally, the interaction dynamics of two and three dark-managed solitons has been numerically explored.

Three distinct and impressive visions for the soliton solutions to the higher-order nonlinear Schrodinger equation

In this article we will use three different schemas to establish three different visions of the soliton solutions to the higher-order nonlinear Schrodinger equation (HONSE). The first schema is the extended simple equation method (ESEM), while the second one is the Paul-Painlevé approach method (PPAM) and the third one is the Ricatti-Bernolli Sub ODE method (RBSOM). The three methods are used in the same vein and parallel to construct many new soliton solutions to this model. A comparison between the obtained results and that achieved previously has been demonstrated.

Propagation properties of chirped optical similaritons with dual-power law nonlinearity

Abstract We study the transmission of femtosecond light waves in an inhomogeneous dual-power law nonlinear model under the influence of self-frequency shift effect. Pulse evolution in such system is described by a generalized higher-order nonlinear Schrodinger equation incorporating the contributions of distributed self-frequency shift, dispersion, dual-power law nonlinearity, and gain/loss. As a result, a variety of analytical similaritons are derived for the first time by a similarity transformation connected to the related constant-coefficients model. We found that in the presence of self-frequency shift effect, the optical similaritons acquire a nonlinear chirp which has a power-law dependence on the intensity of the pulse. The evolution behaviors of presented chirped similaritons are also presented in two different soliton control systems.

The pulses propagation beyond ultra-short range in the systems of optical communication via higher-order nonlinear Schrödinger equation with derivative non-Kerr nonlinear terms

The higher-order nonlinear Schrodinger equation with derivative non-Kerr nonlinear terms is studied in this work. The pulses propagation beyond ultra-short range in the systems of optical communication is described through this dynamical model. The modified extended direct algebraic technique is utilized for deriving different kinds exact solutions such as bright–dark solitons, kink and anti-kink solitons, solitary waves, periodic solutions and elliptic function solutions of this dynamical model. These achieved exact solutions are more general and are useful to researcher for understanding physical phenomena. The structures of some solutions at suitable values of parameters are presented and helpful for knowing the physical interpretation. Several solutions are novel by comparing these solutions with existing solutions. The power and effectiveness of current can observed form constructed solutions.

KdV type asymptotics for solutions to higher-order nonlinear Schrodinger equations

We consider the Cauchy problem for the higher-order nonlinear Schrodinger equation $$\displaylines{ i\partial_t u-\frac{a}{3}| \partial_x| ^3u-\frac{b}{4}\partial_x^4u =\lambda i\partial_x(| u|^2u),\quad (t,x) \in\mathbb{R}^{+}\times \mathbb{R},\cr u(0,x) =u_0(x),\quad x\in\mathbb{R}, }$$ where \(a,b>0\), \(| \partial_x| ^{\alpha}=\mathcal{F}^{-1}| \xi| ^{\alpha}\mathcal{F}\) and \(\mathcal{F}\) is the Fourier transformation. Our purpose is to study the large time behavior of the solutions under the non-zero mass condition \(\int u_0(x)\,dx\neq 0\).
 For more information see https://ejde.math.txstate.edu/Volumes/2020/77/abstr.html

Study on characteristics of internal solitary waves in the Malacca Strait based on Sentinel-1 and GF-3 satellite SAR data

The study of the characteristics of internal solitary waves happened in the Malacca Strait is an urgent problem for submarine, ship navigation and marine engineering. Based on SAR remote sensing data obtained from the high spatial resolution Sentinel-1 and GF-3, the internal solitary wave characteristics of the Malacca Strait are investigated. By use of 20 Sentinel-1 SAR images from June 2015 to December 2016 and 24 GF-3 images from April 2018 to March 2019, the spatial distribution characteristics of internal solitary wave s are statistically analyzed. It is found that the internal solitary waves are usually in the form of wave packets and single solitary waves, and the maximum crest length of the leading wave can reach 39 km. The amplitude and group velocity of internal solitary wave s can be calculated by the inversion model of high-order nonlinear Schrodinger (NLS) equation, and the calculated amplitude of the internal solitary wave s and the propagation group velocity of the wave packets range from 4.7 m to 23.9 m and 0.12 m/s to 0.40 m/s, respectively. The range of phase velocity of single internal solitary waves obtained by KdV equation is 0.26 m/s to 0.60 m/s. In general, the amplitude and the velocity of internal solitary wave s in Malacca strait are related to the topography.

Mixed-type vector solitons for the (2 + 1)-dimensional coupled higher-order nonlinear Schrödinger equations in optical fibers

In this paper, the (2 + 1)-dimensional coupled higher-order nonlinear Schrodinger equations are under investigation. The mixed-type vector solitons are derived via Hirota method. The interactions are analyzed graphically mainly on x–t and x–y planes. Moreover, a high intensity pulse, combined solitary waves and bound state solitons are discovered during the interactions. These results are available in optical fiber communication systems.

Periodic soliton interactions for higher-order nonlinear Schrödinger equation in optical fibers

Research on the interactions between optical solitons is of great significance for the large capacity and long distance transmission in optical fibers. In this paper, interactions between periodic solitons are investigated for the first time. Analytic solution for the high-order nonlinear Schrodinger equation, which can be used to describe the periodic soliton transmission, is obtained based on the bilinear method. Interaction characteristics between periodic solitons are discussed. Influences of corresponding dispersion and nonlinear effects on their interactions are analyzed. Results provide theoretical guidance for the stable transmission of periodic solitons in inhomogeneous optical fibers.

Exact traveling wave solutions to the higher-order nonlinear Schrödinger equation having Kerr nonlinearity form using two strategic integrations.

In this paper, exact traveling wave solutions to the higher-order nonlinear Schrodinger equation having Kerr nonlinearity form are derived, by adopting two relevant architecture of integration methods namely the extended sinh-Gordon equation expansion method and extended Jacobi elliptic function methods (EJEF method). The studied model, describes the propagation of optical solitons in nonlinear optical fibers. As a results, various types of traveling wave solutions are obtained including Jacobi elliptic function solutions. For some special cases, when the modulus of m approach 0 or 1 it is gained respectively periodic wave solutions and hyperbolic function solutions. Comparing our results with the well-known results in literature are also reported. Lastly the 3D profile to some of the obtained solutions are also plotted.

Employing the dynamics of poles in the complex plane to describe properties of rogue waves: case studies using the Boussinesq and complex modified Korteweg–de Vries equations

The dynamics and properties of rogue waves of two classical evolution equations are studied in terms of trajectories of the poles of the exact solutions, by analytically continuing the spatial variable to be complex. The Boussinesq equation describes the motion of hydrodynamic waves in two opposite directions in the shallow water regime. The complex modified Korteweg–de Vries equation is relevant for wave packets in nonlinear media governed by a higher-order nonlinear Schrodinger equation, for the special case where second-order dispersion and cubic self-interaction are absent. On examining the movement of poles of the exact solutions for rogue waves, the real parts of the poles correlate well with the locations of maximum displacements in the physical space. This phenomenon holds for high precision numerically for the first-, second- and third-order rogue waves of the Boussinesq equation. A similar principle is also valid for the first- and second-order rogue waves of the complex modified Korteweg–de Vries equation. The imaginary parts of the poles can generate useful information too. For these two evolution models, a smaller imaginary part in the complex plane is associated with a larger amplitude of the rogue wave in physical space. An empirical formula is proposed which works well for the three lowest orders of rogue waves of the Boussinesq equation.

Dark soliton molecules in nonlinear optics

The study on soliton molecules is one of the important topics in nonlinear science especially in nonlinear optics. The bright soliton molecules have been experimentally observed in optics, however, the dark soliton molecules have not yet been experimentally observed. Theoretically, the soliton molecules have been found for some coupled nonlinear systems. Nevertheless, the soliton molecules have not been obtained for non-coupled single component nonlinear models. In this paper, we first study the exact periodic waves (soliton lattices) and solitary waves for a nonlinear nonintegrable optical model with second and third order dispersions and high order nonlinear effects including self-steeping, Raman scattering and nonlinear dispersion. Two types of dark soliton lattice and three types of soliton lattice are explicitly exhibited for general nonintegrable system. Five types of bright (with and without gray background), dark and gray solitons can be obtained from the limit cases of the modules of the soliton lattices. For an integrable case, using a novel generalized bilinear form of a single component nonlinear system, the multi-soliton solutions are obtained and expressed by a completely new form which are invariant under the full reversal transformations such as the parity, the time reversal, the charge conjugate and the field reversal. To find soliton molecules, a novel mechanism, the velocity resonant, is proposed. Starting from the multi-soliton solutions and using the velocity resonance mechanism, the analytical expression of the dark soliton molecules can be readily obtained. For the model given in this paper, the integrable higher order nonlinear Schrodinger equation, one can proved that the interactions among the dark soliton molecules and the usually solitons are elastic. It is worth pointing out that soliton molecules can also exist in the case of nonintegrable systems.

Finite difference scheme for a higher order nonlinear Schrödinger equation

In this work we present a finite difference scheme used to solve a higher order nonlinear Schrodinger equation. These equations are related to models of propagation of solitons travelling in fiber optics. The scheme is designed to preserve the numerical $$L^2$$ norm, and control the energy for a suitable choose on the equation’s parameters. We prove the convergence of the numerical solution for this equation in a boundary domain, and we show numerical results displaying conservation properties of the schemes using solitons as initial conditions.

Solitons and elliptic function solutions of higher-order dispersive and perturbed nonlinear Schrödinger equations with the power-law nonlinearities in non-Kerr medium

In this paper, the fourth-order dispersive non-linear Schrodinger equation (NLSE) with dual power law nonlinearity and perturbed NLSE with power law nonlinearity in non-Kerr medium are analyzed by employing the improved auxiliary equation technique. The perturbed NLSE depict the quantic nonlinearity effects on promulgation of the ultra-short optical pulses in a non-Kerr medium like an optical fiber. We achieved various types of soliton and elliptic function solutions, some of them are novel and did not exist previously. Graphically, representations of some obtained exact solutions are also given via assigning suitable values to the parameters that aid for understanding the physical phenomenon of these equations. The less computational work and the achieved solutions show that the current proposed technique is powerful and effective. Furthermore many other such types of higher-order NLSEs can be solved using the current method.

A high-order nonlinear Schrödinger equation as a variational problem for the averaged Lagrangian of the nonlinear Klein–Gordon equation

We use Whitham’s averaged Lagrangian method extended with the multiple-scale formalism to derive a sixth-order nonlinear Schrodinger equation for the complex amplitude of the envelope of the slowly modulated wave trains whose evolution is governed by the nonlinear Klein–Gordon equation with polynomial nonlinearity. Such a high-order nonlinear Schrodinger equation is obtained as a variational problem for the Lagrangian averaged over the rapid oscillations of the carrier wave train. As compared to classical Whitham’s approach, we take into account the derivatives of the complex amplitude with respect to slow timescales and long coordinate in the averaged Lagrangian. The coefficients of the high-order nonlinear Schrodinger equation derived in this work identically coincide with those derived by us earlier by the method of multiple scales applied directly to the original nonlinear Klein–Gordon equation. Finally, we consider the sine-Gordon equation as a partial case of the nonlinear Klein–Gordon equation and demonstrate one example of the numerical solution to the corresponding sixth-order nonlinear Schrodinger equation in the form of the envelope quasi-soliton.

Rogue Waves and Their Dynamics on Bright-Dark Soliton Background of the Coupled Higher Order Nonlinear Schrödinger Equation

The coupled higher order nonlinear Schrodinger equation is investigated by using the Darboux-dressing transformation with the Lax pair and asymptotic expansion method. Based on the Lax pair, we fir...

Finite difference scheme for a high order nonlinear Schrodinger equation with localized damping

In this work we present a finite difference scheme used to solve a High order Nonlinear Schrodinger Equation with localized damping. These equations can model the propagation of solitons travelling in fiber optics ([3], [10]). The scheme is designed to preserve the numerical en- ergy at L 2 level, and control the energy at H 1 level for a suitable choose on the equation’s parameters, and when there is no damping in effect. Numerical results will be shown.

Study of soliton solutions of higher-order nonlinear Schrödinger dynamical model with derivative non-Kerr nonlinear terms and modulation instability analysis

Abstract In this work, soliton solutions of higher order non-linear Schrodinger equation (NLSE) with derivative non-Kerr non-linear terms are constructed by employing two analytical techniques, namely, exp ( - Φ ( ζ ) ) -expansion and improved simple equation methods. This dynamical model plays a key role in engineering and physics, and it can be equation of pulses promulgation behind ultra-short range in the system of optical communication. The constructed soliton solution helps researchers for understanding the physical phenomenon of this equation. The standard linear stability analysis is utilized and the stability of model is investigated which substantiate that all results are stable and exact. Graphically, the movements of some solutions are depicted at appropriate values of parameters. The achieved results shows simplicity, reliability and power of the current schemes.