Resonant Nonlinear Schrodinger Equation Research Articles

Bifurcation study, phase portraits and optical solitons of dual-mode resonant nonlinear Schrodinger dynamical equation with Kerr law non-linearity

This study investigates the dynamic characteristics of the dual-mode resonant non-linear Schrodinger equation with a Bhom potential. Hydrodynamics, nonlinear optical fibre communication, elastic media, and plasma physics are just a few of the mathematical physics and engineering applications for this model. The study aims to achieve two main objectives: first, to discuss bifurcation analysis, and second, to extract optical soliton solutions using the extended hyperbolic function method. The study successfully derives various wave solutions, including bright, singular, periodic singular and dark solitons, based on the governing model. The findings conferred in this article show a crucial advancement in understanding the propagation of waves in non-linear media. Additionally, bifurcation of phase portraits of ordinary differential equation consistent with the partial differential equation under consideration is conducted. We also highlight specific constraint conditions that ensure the presence of these obtained solutions. The existing literature shows that these methods are first time applied on this model.

Interactions of pulses produced by two- mode resonant nonlinear Schrodinger equations

Single resonant nonlinear Schrodinger equation RNLSE has wide applications in sciences. It describes the transient state between self-focusing and self-defocusing polarization. This motivated researchers to study and investigate the physical characteristics behind. Here, we are concerned with analyzing the solutions of two-mode RNLSE which may reveal complex phenomena. Novel shapes of pulses propagation in optical fibers are shown Further, the colliding dynamics of waves are inspected. The different characteristics of pulses are defined and interpreted. These features are studied via finding the exact solutions of the two modes RNLSE. These solutions are obtained. by using the unified method. It is found that the criteria of the polarization of the two modes may be, mutual, or of the same polarization. Which depends on the crucial values of the coefficients of the quantum potential. Also, it is shown that the propagation of pulses exhibits multiple-geometric structures.Which are complex chirped, M-W-shaped pulses, rhombus (diamond) and tun able conoidal pulses. These are novel features of pulses propagation.The spectral characteristics show a variety of some important results. Here, it is inspected that the collision is elastic.

Chirped self-similar cnoidal waves and similaritons in an inhomogeneous optical medium with resonant nonlinearity

Abstract Under investigation in this work is the self-similar propagation of light waves inside optical fibers with varying resonant and Kerr law nonlinearities, dispersion, and gain or absorption. A generalized resonant nonlinear Schrodinger equation (NLSE) with distributed parameters that describes the pulse evolution in such inhomogeneous optical system has been studied by the similarity transformation method, and a class of self-similar cnodial wave and similariton solutions is presented. It is found that the obtained bright and dark similaritons have a quadratic phase variation in time implying that the associated frequency chirp is linear. We also discuss the applicability of these linearly chirped optical similaritons in a periodic distributed fiber medium. The results show that a suitable choice of the dispersion and cubic nonlinearity parameters enables one to control the similariton structure and dynamical behavior.

On the Lagrangian version of the Korteweg capillarity system: integrability aspects

A Lagrangian version of the classical 1+1-dimensional Korteweg capillarity system is shown to encapsulate as a particular reduction an integrable Boussinesq equation. The associated integrability of the corresponding Eulerian capillarity system is set down. In turn, hidden integrability cases of the generalised Boussinesq equation are recorded associated with reduction of the Eulerian system to the integrable cubic and resonant nonlinear Schrodinger equations together with extensions obtained by application of invariance under a reciprocal transformation.

Stochastic treatment of the solutions for the resonant nonlinear Schrödinger equation with spatio-temporal dispersions and inter-modal using beta distribution

In this paper, the extended Jacobian elliptic function expansion method is implemented in order to construct some new traveling wave solutions for the resonant nonlinear Schrodinger equation with both spatio-temporal dispersion and inter-modal dispersion. These new traveling wave solutions are obtained by the proposed method, which is easy to implement and computationally very attractive. Moreover, these solutions may be applicable for some physical fields, such as plasma physics. The main aim of this paper is the stochastic treatment of the solutions when the spatio-temporal coefficient or the wave transition is beta random variables. The priority of using beta statistical distribution for the spatio-temporal is discussed. Some graphical simulations are given to illustrate the behavior of these solutions in the deterministic and stochastic case studies. Indeed the proposed techniques are very powerful tool to solve other models in applied science.

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.

A novel technique to construct exact solutions for nonlinear partial differential equations

The aim of the manuscript is to present a new exact solver of nonlinear partial differential equations. The proposed technique is developed by extending the $ \phi^{6}$-model expansion method as a known method. The corresponding exact solutions are given in terms of Jacobi elliptic functions. Some new optical solutions of the resonant nonlinear Schrodinger equation are constructed within this newly proposed method. For some specific choices of the modulus of Jacobi elliptic functions, various solutions of the equation are introduced. Some numerical simulations are also included to emphasize that all parameters have major influences for the solitary waves behaviours. The proposed technique is very simple and straightforward, and can be employed to solve other non-linear partial differential equations.

Generalized optical soliton solutions to the (3+1)-dimensional resonant nonlinear Schrödinger equation with Kerr and parabolic law nonlinearities

The resonant nonlinear Schrodinger equation in three dimensions featuring both the Kerr and parabolic law nonlinearities has been considered in this paper due to its vital role in the propagation of solitons in optical fibers. Optical soliton solutions to this important model consisting of dark, singular and combo solitons alongside various periodic solutions that are new and more general to the lately devised solutions have been presented. We also reported the constraint conditions that guarantee the survival of these solitons and depicted certain results.

Analytical study of resonant optical solitons with variable coefficients in Kerr and non-Kerr law media

In this article, the extended trial approach has been used for the investigation of traveling wave solutions of resonant nonlinear Schrodinger equation with variable coefficients depending upon time in the presence of different effects such as Kerr law and parabolic law nonlinearity including a Bohm potential term. Many new soliton solutions such as bright, dark, singular and rational function solutions are obtained. This technique proves to be an advantageous and powerful mathematical tool for establishing abundant exact traveling wave solutions in mathematical physics and plasma. The resonant nonlinear Schrodinger’s equation, in specific, governs the propagation of soliton in both quantum potential and uniaxial waves in a cold collisionless plasma.

New $\phi^{6}$ ϕ 6 -model expansion method and its applications to the resonant nonlinear Schrödinger equation with parabolic law nonlinearity

With the aid of symbolic computation, the new $\phi^{6}$ -model expansion method is applied, in this article, for the first time to the resonant nonlinear Schrodinger equation with parabolic law nonlinearity to find families of Jacobi elliptic function solutions. In particular, when the modulus of the Jacobi elliptic functions tends to one or to zero, we can get the hyperbolic and trigonometric function solutions, respectively. This new method presents a wider applicability for handling the nonlinear partial differential equations. Comparison of our new results with the well-known results are given. At the end of this paper, we use the solutions of the Lienard equation to find more different solutions for the proposed resonant nonlinear Schrodinger equation mentioned above.

Optical soliton perturbation for time fractional resonant nonlinear Schrödinger equation with competing weakly nonlocal and full nonlinearity

In this article, we retrieve optical soliton solutions of the perturbed time fractional resonant nonlinear Schrodinger equation having competing weakly nonlocal and full nonlinearity. We study the equation for two different forms of nonlinearity, namely Kerr law and anti-cubic law. The F-expansion method along with fractional complex transformation is used to obtain the optical solitons. Moreover, the existence of these solitons are guaranteed with the constraint relations between the model coefficients and the traveling wave frequency coefficient.

A new generalized exponential rational function method to find exact special solutions for the resonance nonlinear Schrödinger equation

The present paper suggests a novel technique to acquire exact solutions of nonlinear partial differential equations. The main idea of the method is to generalize the exponential rational function method. In order to examine the ability of the method, we consider the resonant nonlinear Schrodinger equation (R-NLSE). Many variants of exact soliton solutions for the equation are derived by the proposed method. Physical interpretations of some obtained solutions is also included. One can easily conclude that the new proposed method is very efficient and finds the exact solutions of the equation in a relatively easy way.

Solitons and other solutions to the resonant nonlinear Schrodinger equation with both spatio-temporal and inter-modal dispersions using different techniques

In this paper, we apply different techniques, namely, the Jacobi elliptic expansion method, the modified simple equation method, the sine–cosine method, the extended auxiliary equation method, the new mapping method and the improved sub-ODE method for constructing many new types of Jacobi elliptic function solutions, solitons and other solutions to the resonant nonlinear Schrodinger equation with both spatio-temporal and inter-modal dispersions. Comparing our new results with the well-known results are given. Also, we compare the results yielding from the used different techniques with each other.

Optical soliton solutions of the generalized higher-order nonlinear Schrödinger equations and their applications

The propagation of the optical solitons is usually governed by the higher order nonlinear Schrodinger equations (NLSE). In optics, the NLSE modelizes light-wave propagation in an optical fiber. In this article, modified extended direct algebraic method with add of symbolic computation is employed to construct bright soliton, dark soliton, periodic solitary wave and elliptic function solutions of two higher order NLSEs such as the resonant NLSE and NLSE with the dual-power law nonlinearity. Realizing the properties of static and dynamic for these kinds of solutions are very important in various many aspects and have important applications. The obtaining results confirm that the current method is powerful and effectiveness which can be employed to other complex problems that arising in mathematical physics.

Optical solitons for the resonant nonlinear Schrödinger equation with competing weakly nonlocal nonlinearity and fractional temporal evolution

This paper is devoted in the study of the resonant nonlinear Schrodinger equation with competing weakly nonlocal nonlinearity and fractional temporal evolution which describes the dynamics of optical solitons through nonlinear optical fibers. The equation is studied with two forms of nonlinearity, namely Kerr law and parabolic law. The fractional complex transformation and F-expansion method is used to obtain exact optical soliton solutions of the equation. In addition, the constraint relations between the model coefficients and the traveling wave frequency coefficient for the existence of optical solitons are also derived.

Abundant soliton solutions of the resonant nonlinear Schrödinger equation with time-dependent coefficients by ITEM and He’s semi-inverse method

The improved $$\tan (\phi /2)$$ -expansion method (ITEM) and He’s semi-inverse variational method (HSIVM) are the efficient methods for obtaining exact solutions of nonlinear differential equations. In this paper, the ITEM and HSIVM are applied to construct exact solutions of the resonant nonlinear Schrodinger equation (RNLSE) with time-dependent coefficients for parabolic law nonlinearity. The resonant nonlinear Schrodinger equation plays a very important role in mathematical physics and nonlinear optics. We compare analytical findings with the results of the other analytical schemes describing the ansatz method approach and expansion method are used to carry out the integration. Description of the ITEM is given and the obtained results reveal that the ITEM is a new significant method for exploring nonlinear partial differential models. Moreover, by help of the HSIVM we obtained the bright and dark soliton wave solutions. Finally, by using Matlab, some graphical simulations were drawn to see the behavior of these solutions.

New methods to solve the resonant nonlinear Schrödinger’s equation with time-dependent coefficients

In this study, the generalized $$\tan (\phi /2)$$ -expansion method and He’s semi-inverse variational method (HSIVM) are applied to seek the exact solitary wave solutions for the resonant nonlinear Schrodinger equation with time-dependent coefficients. Using these methods, we investigate exact solutions for the nonlinear resonant Schrodinger equation with time-dependent coefficients two forms of nonlinearity, including power and dual-power law nonlinearity. Moreover, many new analytical exact solutions are obtained which are expressed by hyperbolic solutions, trigonometric solutions, and rational solutions. In addition, we obtained the bright soliton by HSIVM. These methods are powerful, efficient and those can be used as an alternative to establishing new solutions of different types of differential equations in mathematical physics and engineering.

Kerr-law nonlinearity of the resonant nonlinear Schrodinger’s equation with time-dependent coefficients

This work deals with exact soliton solutions of the resonant nonlinear Schrodinger equation with time-dependent coefficients. The propagation equation that is the resonant dispersive nonlinear Schrodinger’s equation with Kerr law nonlinearity is studied by two analytical methods, namely, the improved tan( $$\phi /2$$ )-expansion method (ITEM) and He’s semi-inverse variational method (HSIVM), based upon the integration tools. We compare analytical findings with the results of the other analytical schemes describing the ansatz method approach and expansion method are used to carry out the integration. Description of the ITEM is given and the obtained results reveal that the ITEM is a new significant method for exploring nonlinear partial differential models. Moreover, by help the HSIVM we obtained the bright soliton wave solution. Finally, by using Matlab, some graphical simulations were done to see the behavior of these solutions.

Non topological 1-soliton solution to resonant nonlinear schrodinger equation with kerr law nonlinearity

In this paper, using the methods of ansatz, sine-cosine and He’s semi-inverse variation, non-topological 1-soliton solution to Resonant Nonlinear Schrodinger Equation with Kerr law nonlinearity is obtained. The results show that these methods are very effective ones for finding exact solutions to various types of nonlinear evolution equations appearing in the studies of science and engineering.

Jacobi elliptic solutions, soliton solutions and other solutions to four higher-order nonlinear Schrodinger equations using two mathematical methods

Two mathematical methods, via the (G′/G)-expansion method and extended auxiliary equation method are applied in this article to find many exact solutions with parameters of four higher-order nonlinear Schrodinger equations whose balance numbers are not positive integers, namely, the nonlinear Schrodinger equation with dual power-law nonlinearity, the higher-order dispersive nonlinear Schrodinger equation with both fourth-order dispersion effects and a quintic nonlinearity, the resonant nonlinear Schrodinger equation and the generalized higher-order nonlinear Schrodinger equation. When the parameters take up special values, the solitary and singular solitary wave solutions of these nonlinear Schrodinger equations are given. The used methods in this article present a wider applicability for handling nonlinear wave equations. Comparing our new results and the well-known results are obtained. Comparing the results resulting from the two methods with each others are also given.