Dimensional Nonlinear Schrodinger Equation Research Articles

Travelling Wave-like Solutions for Some Nonlinear Equations with Cubic Nonlinearities

We exercised the series-expansion method to extract solitary wave solutions for complex nonlinear evolution equations (NLEEs) such as the coupled Higgs equation (CHE), the (3+1)-dimensional dynamical system, the (3+1)-dimensional nonlinear Schrodinger(NLSE) equation which have many physical importance in different branches. Varieties of periodic and solitonic wave-like solutions are extracted. Computational work has been done and plots and counter graphs are plotted using Wolfram Mathematica 11.

The weakly non-linear waves propagation for Kelvin-Helmholtz instability in the magnetohydrodynamics flow impelled by fractional theory.

The weakly nonlinear wave propagation that occurs in the presence of magnetic fields, in which energy is concentrated in a narrow band of wave-numbers in a dispersive and dissipative fluid. The main objective of this paper is to analyze the dimensional elliptic nonlinear Schrodinger equation under the influence of three different fractional operators. The generalized fractional soliton solutions and propagation of magnetohydrodynamics fluid in sort of solition will be visualized. The Conformable, and M-truncated fractional operator applied to classical evolution Schrodinger equation. In order to get the analytical closed form solution, one of the generalized approach new extended direct algebraic method is utilized. The fractional nonlinear elliptic Schrodinger equation is developed in three different fractional sense. The similarity transformation technique converted the controlling fractional system to ordinary differential equations. The fractional analytical solutions such as, plane solution, mixed hyperbolic solution, periodic and mixed periodic solutions, mixed trigonometric solution, trigonometric solution, shock solution, mixed shock singular solution, mixed singular solution, complex solitary shock solution, singular solution and shock wave solutions are obtained. The graphical 2-D and 3-D representation of the results is shown to express the propagation of fluid with the magnetic field by assuming the appropriate values of the involved parameters. The graphical performance of the obtained solution at various settings of parametric values and fractional order reveals new perspectives and fascinating model phenomena. The attained outcomes have significant applications and have opened up innovative development areas for research across numerous scientific fields.

A KAM theorem for higher dimensional reversible nonlinear Schrodinger equations

In this article we prove an abstract Kolmogorov-Arnold-Moser (KAM) theorem for infinite dimensional reversible systems. Using this theorem, we obtain the existence of quasi-periodic solutions for a class of reversible (non-Hamiltonian) coupled nonlinear Schrodinger systems on a d-torus.

The study of new optical soliton solutions to the time-space fractional nonlinear dynamical model with novel mechanisms

In this work, we successfully employ two constructive mathematical tools such as the extended rational sine-cosine/sinh-cosh and novel Φ6-expansion model techniques, for obtaining different types of optical solitons and other solutions to the (2+1)-dimensional time-space fractional nonlinear Schrodinger (NLS) equation within the modified Riemann–Liouville derivatives. Solitary wave, dark, singular, and their combined solutions are retrieved. Moreover, hyperbolic, trigonometric, plane wave periodic wave, and Jacobi elliptic function solutions are also derived. The novelty of the obtained solutions is revealed by comparing them with the previous finding. By selecting suitable parametric values, numerical simulation and physical interpretations of the achieved outcomes are manifested with interesting figures presenting the meaning of the acquired results. The novel approaches are used through the symbolic computations impart a dynamical and potent mathematical implement for solving various benevolent nonlinear wave problems. The results suggest that these constructing techniques are concise, direct, efficient, and straightforward. The resulting solutions are novel, intriguing, and potentially helpful for understanding energy transit and diffusion processes in mathematical models of several disciplines of interest, including telecommunication, engineering, mathematical biology, mathematical physics, and nonlinear optics etc.

Assorted soliton structures of solutions for fractional nonlinear Schrodinger types evolution equations

Fractional order nonlinear evolution equations have emerged in recent times as being very important model for depicting the interior behavior of nonlinear phenomena that exist in the real world. In particular, Schrodinger-type fractional nonlinear evolution equations constitute an aspect of the field of quantum mechanics. In this study, the (2 + 1)-dimensional time-fractional nonlinear Schrodinger equation and (1 + 1)-dimensional time-space fractional nonlinear Schrodinger equation are revealed as having different and novel wave structures. This is shown by constructing appropriate analytic wave solutions. A successful implementation of the advised rational (1/ϕ′(ξ))-expansion method generates new outcomes of the considered equations, by comparing them with those already noted in the literature. On the basis of the conformable fractional derivative, a composite wave variable conversion has been used to adapt the suggested equations into the differential equations with a single independent variable before applying the scheme. Finally, the well-furnished outcomes are plotted in different 3D and 2D profiles for the purpose of illustrating various physical characteristics of wave structures. The employed technique is competent, productive and concise enough, making it feasible for future studies.

Plane Wave Solutions With Dynamical Behaviors for Heisenberg Model of Ferromagnetic Spin Chain With Beta Derivative Evolution and Obliqueness

The plane waves with their dynamical behaviors of (2+1)-dimensional nonlinear Schrodinger equation (NLSE) having beta derivative spatial-temporal evolution are investigated. In order to study such phenomena, NLSE is converted to a nonlinear ordinary differential equation with a planar dynamical system (PDS) by considering the variable wave transform and the properties of the beta derivative. By employing two distinct solution methods, namely, the auxiliary ordinary differential equation method and the extended simplest equation method, some more new general form of analytical solutions of NLSE are constructed. The effect of beta derivative parameter and obliqueness on several types of wave structures along with the phase portrait diagrams are reported. It is found that the PDS is not supported by any type of orbits for Θ = 45°. It is also confirmed from the obtained solutions that no plane waves are generated for Θ = 45°. The presented studies on bifurcation analysis and analytical solutions for (2+1)-dimensional NLSE would be very useful to understand the physical scenarios for Heisenberg models of ferromagnetic spin chain with magnetic interactions.

Hybrid behavior of a two-dimensional Noguchi nonlinear electrical network

In this work, the propagation of modulated waves in a nonlinear two-dimensional discrete electrical line is studied. Based on the linear dispersive law, we demonstrate that the network can adopt Hybrid’s behavior due to the fact that, without changing its appearance structure or its parameters values, it can become alternatively, a purely right-handed line, a purely left-handed line or a composite right-left-handed line depending on coupling transverse parameters and Using the reductive perturbation method, a (2+1)-dimensional nonlinear Schrodinger equation governing the dynamics of the small amplitude signals in the network is derived and the impact of the above transverse parameters on the sign of the product of their coefficients is examined. Likewise, backward and forward solitary wave propagations in the system is predicted and analyzed quantitatively and qualitatively. The effects of relevant transverse network parameters on the characteristic parameters of these waves are investigated. We find out that these transverse parameters considerably affect the characteristics and the nature of the waves that are propagated throughout the system.

Rogue wave signals in a coupled anharmonic network: effects of the transverse direction

In this paper, we consider the dynamics of modulated waves in a nonlinear two-dimensional discrete electrical lattice made of several well-known anharmonic modified Noguchi lines coupled transversely to one another with a linear inductor. In the linear limit and depending on the chosen frequency domain, we demonstrate that the network behaves either as a right-handed line or as a left-handed line. We also establish that the band pass character of the modified Noguchi line increases the bandwidth of the allowed frequencies for which the system adopts the left-handed behavior. Based on the reductive perturbation method in the semi-discrete approximation, we derive a (2 + 1)-dimensional nonlinear Schrodinger (2D-NLS) equation governing the dynamics of the small amplitude signals in the system. The specific nature of this amplitude equation is examined, and it appears that its hyperbolic or elliptic character strongly depends on the coupling inductor $$L_{2}$$ . Moreover, using a simple transformation, we reduce the obtained equation to a (1 + 1)-dimensional nonlinear Schrodinger equation (1D-NLS). The exact first- and second-order rogue wave’s solutions of this 2D-NLS equation are derived in bandwidth frequencies where the network may exhibit modulational instability. The results of our investigations display number of interesting rogue wave profiles which have not been observed in the scientific literature. Furthermore, we examine the effects of both the coupling linear inductor $$L_{2}$$ and the nonlinear dispersion factor $$\eta$$ on the characteristics of the rogue waves and also on their propagation through the network. It appears that $$L_{2}$$ and $$\eta$$ can be used to squeeze or stretch the shape of the rogue waves signals traveling in the network. These numerical results are in perfect agreement with the analytical predictions.

Dynamic behaviors for a (2 + 1)-dimensional inhomogenous Heisenberg ferromagnetic spin chain system

We investigate a (2 + 1)-dimensional nonlinear Schrodinger equation (NLSE), which describes the spin dynamics of (2 + 1)-dimensional inhomogeneous Heisenberg ferromagnetic spin chain (IHFSC) with bilinear and anisotropic interactions in the semiclassical limit. Miscellaneous new solitons solutions are obtained through the generalized Riccati equation mapping method (GREMM). Moreover, the effects of homogeneity on the soliton propagation and interaction are discussed. The derived structure of the obtain solutions offers a rich platform to better understand the nonlinear dynamics in the ferromagnetic materials.

New solitons and conditional stability to the high dispersive nonlinear Schrödinger equation with parabolic law nonlinearity

The complete discrimination system was proposed to search traveling wave solutions for a (1 + 1)-dimensional nonlinear Schrodinger equation with parabolic law nonlinearity. As a consequence, we obtained a series of solutions which contain dark soliton, bright soliton, triangular solitary wave solution, kink soliton, periodic solutions and implicit analytical solutions. At the same time, numerical simulations were demonstrated to visualize the mechanics of these solutions. Lastly, linear stability of dark soliton $$u_{25}$$ in Lyapunov’s sense was discussed.

Optical solitons for the fractional $$(3+1)$$-dimensional NLSE with power law nonlinearities by using conformable derivatives

In this paper, the process of the extended direct algebraic method (EDAM) is used to obtain the optical solitons in fractional (3 + 1)-dimensional nonlinear Schrodinger equation through the conformable derivative. Firstly, this fractional equation is changed into the ordinary differential equation by using the wave variables transformation. Then, new several forms of optical solitons are obtained by using EDAM.

Managements of scalar and vector rogue waves in a partially nonlocal nonlinear medium with linear and harmonic potentials

We consider a ( $$2+1$$ )-dimensional nonautonomous-coupled nonlinear Schrodinger equation, which includes the partially nonlocal nonlinearity under linear and harmonic potentials. Via a projecting expression between nonautonomous and autonomous equations, and utilizing the bilinear method and Darboux transformation method, we find diversified exact solutions. These solutions contain the nonlocal rogue wave and Akhmediev or Ma breather solutions, and the combined solution which describes a rogue wave superposed on an Akhmediev or Ma breather. By adjusting values of diffraction, width and phase chirp parameters of wave, the maximum value of the accumulated time can be modulated. When we compare the maximum value of the accumulated time with that of the excitation position parameters, we study the management of scalar and vector rogue waves, such as the excitations of full shape, early shape and climax shape for rogue waves.

The weakly nonlinear wave propagation theory for the Kelvin-Helmholtz instability in magnetohydrodynamics flows

Abstract In this paper, the weakly nonlinear wave propagation theory in the occurrence of magnetic fields in fluids of superposed is studied. The amplitude evolution of governing equations of the amplitude of the progressive waves is reported. The soliton and other kinds solutions of (2+1)-dimensional elliptic nonlinear Schrodinger equation is constructed. We also investigate the properties of envelope of the (2+1)-dimensional wave pocket. The propagation of pulses ahead of ultra-short range in the optical communication systems is illustrated by this dynamical equation. The modified and extended rational expansion method is utilized for obtaining different kinds of wave solutions such as bright-dark solitons, solitons of Kink and anti-Kink, solitary waves, periodic solutions and elliptic function solutions of this equation. These achieved exact solutions are further general and useful to researcher for understanding physical phenomena. Several solutions are novel by comparing these solutions with existing solutions. The power and effectiveness of current can observed form constructed solutions.

$$\mathbf{L^2}$$-decay rate for the critical nonlinear Schrödinger equation with a small smooth data

We consider the Cauchy problem for the one dimensional nonlinear dissipative Schrodinger equation with a cubic nonlinearity $$\lambda |u|^2u$$, where $$\lambda \in {\mathbb {C}}$$ with Im $$\lambda < 0$$. We show that a relation between $$L^2$$-decay rate for the solution and a smoothness of the initial data. Our result improves the recent work of Hayashi–Li–Naumkin (Adv Math Phys Art. ID 3702738, 7, 2016) for the decay rate of $$L^2$$.

Oblique plane waves with bifurcation behaviors and chaotic motion for resonant nonlinear Schrodinger equations having fractional temporal evolution

This study deal with the oblique plane wave solutions with dynamical behaviours for (2 + 1)-dimensional resonant nonlinear Schrodinger equations having Bhom’s quantum potential with distinct law of nonlinearities (Kerr and parabolic law) and fractional temporal evolution. The considered equations are converted to solvable form by assuming conformable Khalil’s fractional derivatives. The bifurcation behaviors and chaotic motion for the existence of traveling waves are investigated by forming the planar dynamical system from the considered equations. The novel auxiliary ordinary differential equation method is therefore used to divulge several forms of plane wave solutions of these equations. It is investigated that the widths of resonant wave dynamics are significantly modified with the influence of obliqueness. The evaluated results may be very useful for better examining the resonant optical solitons in nonlinear dynamics because of obliqueness are existed in various nonlinear systems, specifically in optical bullets, Madelung fluids, etc.

Three-dimensional Nonlinear Structures in Magnetized Complex Plasmas

A three-dimensional multi-component magneto-plasma system consisting of hybrid nonextensive-nonthermal-distributed electrons, positive ions, and immobile negatively charged dust grains is considered to examine the modulational instability (MI) of dust ion-acoustic waves (DIAWs). The reductive perturbation method, which is valid for small but finite amplitude DIAWs, is employed to derive the (3 + 1)-dimensional nonlinear Schrodinger equation (NLSE). The NLSE leads to the MI of DIAWs as well as the formation of dust ion-acoustic rogue waves (DIARWs), which are formed due to the effects of nonlinearity in the propagation of DIAWs. It is found that the basic features (viz. amplitude and width) of the DIAWs and DIARWs (which is formed in the unstable region) are significantly modified by the various plasma parameters such as non-extensive (non-thermal) parameter q(α), number densities of plasma species, threshold modulational obliqueness ($${{\theta }_{c}}$$), etc. The application of the results in space and laboratory magneto-plasma system is briefly discussed.

Two-color pulse collapse events in dispersive media

It is well known that an intense pulse sent through a medium with an intensity-dependent index will acquire a stationary shape (spatial soliton) while focusing and collapsing into a point. Many nonlinear phenomena, such as THz generation, require the simultaneous propagation and collapse at two wavelengths. It is shown here that the Kerr effect introduces a coupling between pulses sent co-propagating into a transparent medium with a nonlinear index. Because of this coupling, the pulses at the different wavelengths reshape towards a stationary pair that evolves towards a common focus in time and space. The effect of normal and anomalous dispersion on two-color pulse collapse is investigated numerically. The model to be considered here is an extension of (2+1)-dimensional nonlinear Schrodinger equations (NLSEs) by inclusion of dispersion and for a beam consisting of two frequencies. As such, our study centers on a system of coupled (3+1)-D NLSEs describing the co-propagation of two pulses in the non-resonant regime under self-focusing. We should emphasize that this model gives new insight on the initial dynamics of the two-color filament. While inclusion of the small normal dispersion tends to a temporal split of the beam, anomalous dispersion facilitates collapse. In considering initial “short” (fs) and “long” (ps–ns) temporal pulses, our results present different scenarios of the initial evolution that include the role dispersion may have.

Solitary Wave Solutions for $$(1+2)$$-Dimensional Nonlinear Schrödinger Equation with Dual Power Law Nonlinearity

Here, \(\tan \left( \frac{\phi (\xi )}{2}\right) \)-expansion method is being applied on \((1+2)\)-dimensional nonlinear Schrodinger equation (NLSE) with dual power law nonlinearity. Spatial solitons and optial nonlinearities in materials like photovoltaic–photorefractive, polymer and organic can be identified by seeking help from NLSE with dual power law nonlinearity. Abundant exact traveling wave solutions consisting free parameters are established in terms of exponential functions. Various arbitrary constants obtained in the solutions help us to discuss the graphical behavior of solutions and also grants flexibility to form a link with large variety of physical phenomena. Moreover, graphical representation of solutions are shown vigorously in order to visualize the behavior of the solutions acquired for the equation.

Elliptic Solitons in (1+2)-Dimensional Anisotropic Nonlocal Nonlinear Fractional Schrödinger Equation

This paper obtains the numerical solutions of the elliptic solitons in a (1+2)-dimensional anisotropic nonlocal nonlinear fractional Schrodinger equation, and verifies their stabilities by the direct propagation method. The results show that the properties of such solitons relatively depend on the Levy index. Such as the soliton shape varies with the change of Levy index. When the Levy index decreases, the ellipticity will increase, while the critical power will decrease. Furthermore, we demonstrate the physical features exhibited by the higher order elliptic solitons for a different Levy index.

Periodical collision between hollow solitons in (2+1)-dimensional nonlocal nonlinear Schrödinger equation

Abstract We theoretically investigate the co-propagation characteristics of two hollow solitons in (2+1)-dimensional nonlocal nonlinear Schrodinger equation, which can be regarded as a mathematical model of optical nonlocal nonlinear media. It is found that the two solitons always collide periodically. The influences of initial parameters (such as the initial phase, the input energy, the initial interval etc.) on the evolutions of soliton collision are analyzed in detail. A set of analytical expressions are obtained to describe these characteristics, which are illustrated by the numerical results.