Higher-order Nonlinear Schrodinger Research Articles

The quasi-Gramian solution of a non-commutative extension of the higher-order nonlinear Schrödinger equation

The nonlinear Schrödinger (NLS) equation, which incorporates higher-order dispersive terms, is widely employed in the theoretical analysis of various physical phenomena. In this study, we explore the non-commutative extension of the higher-order NLS equation. We treat real or complex-valued functions, such as g 1 = g 1(x, t) and g 2 = g 2(x, t) as non-commutative, and employ the Lax pair associated with the evolution equation, as in the commutation case. We derive the quasi-Gramian solution of the system by employing a binary Darboux transformation. The soliton solutions are presented explicitly within the framework of quasideterminants. To visually understand the dynamics and solutions in the given example, we also provide simulations illustrating the associated profiles. Moreover, the solution can be used to study the stability of plane waves and to understand the generation of periodic patterns within the context of modulational instability.

NLS-type equations from quadratic pencil of Lax operators: Negative flows

We formulate and study an integrable model of Nonlinear Schrödinger (NLS)-type through its Lax representation, where one of the Lax operators is quadratic and the other has a rational dependence on the spectral parameter. We discuss the associated spectral problem, the Riemann-Hilbert problem formulation, the conserved quantities, as well as a generalisation for symmetric spaces. In addition we explore the possibilities for modelling with higher order NLS (HNLS) integrable equations and in particular, the relevance of the proposed system.

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.

From a generalized discrete NLS equation in discrete alpha helical proteins to the fourth-order NLS equation

In this paper, we consider a generalized integrable discrete nonlinear Schrödinger (NLS) equation, which can describe the dynamics of discrete alpha helical proteins with higher-order excitations and lead to the higher-order NLS equation in the continuum limit. The Darboux transformation (DT) and the soliton solutions of this generalized discrete NLS equation are implemented. It is shown that the integrable properties of the generalized discrete NLS equation, including the discrete Lax pair, the DT and the discrete soliton solutions, give rise to their continuous counterparts as the discrete space step tends to zero.

Inverse scattering and N-triple-pole soliton and breather solutions of the focusing nonlinear Schrödinger hierarchy with nonzero boundary conditions

Based on the matrix Riemann-Hilbert problem we present the inverse scattering transform for the focusing nonlinear Schrödinger (NLS) equation with nonzero boundary conditions (NZBCs) at infinity and triple zeros of analytical scattering coefficients. As a consequence, the inverse scattering problem can be solved via the study of the matrix Riemann-Hilbert problem with triple poles. We find the general N-triple-pole solutions for the potential, and explicit N-triple-pole expression for the reflectionless potential in terms of two determinants. Besides, the trace formulae and theta conditions are also given. Finally, we discuss the one-triple-pole breather-breather-breather solution for the focusing NLS equation with NZBCs. Moreover, the general N-triple-pole breathers of NZBCs can reduce to general N-triple-pole solitons of the zero-boundary conditions. These results can also be extended to the focusing higher-order NLS equations (i.e., the NLS hierarchy such as the Hirota equation, complex mKdV equation, Lakshmanan-Porsezian-Daniel equation, complex fifth-order mKdV equation, fifth-, sixth-, and seventh-order NLS equations, etc.) with NZBCs to find their inverse scattering transforms (ISTs) and N-triple-pole solutions. These obtained triple-pole solutions are useful to explain the related nonlinear wave phenomena.

On the Stabilization of Breather-type Solutions of the Damped Higher Order Nonlinear Schrödinger Equation

Spatially periodic breather solutions (SPBs) of the nonlinear Schrödinger (NLS) equation are frequently used to model rogue waves and are typically unstable. In this paper we study the effects of dissipation and higher order nonlinearities on the stabilization of N-mode SPBs, 1≤N≤3, in the framework of a damped higher order NLS (HONLS) equation. We observe the onset of novel instabilities associated with the development of critical states resulting from symmetry breaking in the damped HONLS system. We develop a broadened Floquet characterization of instabilities of solutions of the NLS equation by showing that instabilities are associated with degenerate complex elements of not only the discrete, but also the continuous Floquet spectrum. As a result, the Floquet criteria for the stabilization of a solution of the damped HONLS centers around the elimination of all complex degenerate elements of the spectrum. For a given initial N-mode SPB, a short-time perturbation analysis shows that the complex double points associated with resonant modes split under the damped HONLS while those associated with nonresonant modes remain closed. The corresponding /damped HONLS numerical experiments corroborate that instabilities associated with nonresonant modes persist on a longer time scale than the instabilities associated with resonant modes.

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.

Dynamics of fundamental solitons and rogue waves on the mixed backgrounds

On the mixed nonzero–zero backgrounds, the vector fundamental breathers, solitons and rogue waves of the coupled higher-order nonlinear Schrodinger system with sign-alternating nonlinearity are investigated. Using the Lax pair and Darboux transformation, we produce a family of vector solutions on the mixed backgrounds. Via the breather-to-soliton conversion, we show the vector solitons on the mixed backgrounds. We show the vector fundamental rogue waves of ultra-high peak amplitude on the mixed backgrounds.

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.

Soliton Solutions of High-order Nonlinear Schrödinger Equations with Different Laws of Nonlinearities

In the present paper, high-order nonlinear Schrodinger equations in non-Kerr law media with different laws of nonlinearities are studied. In this respect, after considering a complex envelope and distinguishing the real and imaginary portions of the models, describing the propagation of solitons through nonlinear optical fibers, their soliton solutions are obtained using the well-organized new Kudryashov method. It is believed that the new Kudryashov method provides an effective mathematical tool to look for soliton solutions of high-order nonlinear Schrodinger equations.

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

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.

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.

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.

Integration Of A Higher-Order Nonlinear SchröDinger System with a Self-Consistent Source in the Class of Periodic Functions

We use the inverse spectral method to integrate a higher-order nonlinear Schrodinger system with a self-consistent source in the class of periodic functions.