Dark Solitary Wave Solutions Research Articles

New soliton solutions to the perturbed nonlinear Schrödinger equation by exp(− Φ(ξ))-expansion method

The perturbed nonlinear Schrödinger equation (PNLSE), used to investigate the dynamics of wave propagation of light in nonlinear optical fibers and planar wave guides is considered in this article. The model is considered in the presence of full nonlinearity and perturbed terms. By using the state-of-the-art integration scheme, exp(− Φ(ξ))-expansion method, different structures of explicit solutions such as dark, singular, rational and periodic solitary wave solutions are celebrated. All the newly found solutions are discussed with their existence criteria.

Exact solitary waves for the 2D Sasa-Satsuma equation

Abstract In this work, we investigate exact solitary wave solutions for the 2D Sasa-Satsuma equation by using the envelope transform and Jacobi elliptic function expansion method. We obtain breather, bright and dark solitary wave solutions. It is shown that 2D Sasa-Satsuma equation has very rich dynamical behavior and can describe the propagation of nonlinear waves in many fields of physics.

Optical solitons for Biswas–Milovic equation by new extended auxiliary equation method

Biswas–Milovic equation is used to narrate the nonlinear dynamics of optical solitons in nonlinear optics. With the aid of new extended auxiliary method, we find Jacobi elliptic functions solutions which degenerate to bright, dark, singular and other solitary wave solutions with Kerr law and dual power law nonlinearities.

Study on the generation mechanism of bright and dark solitary waves and rogue wave for a fourth-order dispersive nonlinear Schrödinger equation

In this paper, we study the generation mechanism of bright and dark solitary waves and rogue wave for the fourth-order dispersive nonlinear Schrödinger (FODNLS) equation, which can not only model the nonlinear propagation and interaction of ultrashort pulses in the high-speed optical fiber transmission system, but also govern the nonlinear spin excitations in the onedimensional isotropic biquadratic Heisenberg ferromagnetic spin with the octupole-dipole interaction. Firstly, via the phase plane analysis, we obtain both the homoclinic and heteroclinic orbits for the two-dimensional plane autonomous system reduced from the FODNLS equation. Further, we derive the bright and dark solitary wave solutions under the corresponding conditions, which reveals the relationship between the homoclinic (heteroclinic) orbit and solitary wave. Secondly, based on the exact first-order breather solution of the FODNLS equation over a nonvanishing background, we give the explicit expressions of group and phase velocities, and reveal that there exists a jump in both the velocities. Finally, in order to verify that the breather becomes a rogue wave at the jumping point, we obtain the first-order rogue wave solution by taking the limit of the breather solution at such point, which confirms the relationship of the generation of rogue wave with the velocity discontinuity.

Analytical methods via bright–dark solitons and solitary wave solutions of the higher-order nonlinear Schrödinger equation with fourth-order dispersion

In this research work, we investigated the higher-order nonlinear Schrödinger equation (NLSE) with fourth-order dispersion, self-steepening, nonlinearity, nonlinear dispersive terms and cubic-quintic terms which is described as the propagation of ultra-short pulses in fiber optics. We apply the modification form of extended auxiliary equation mapping method to find the new exact and solitary wave solutions of higher-order NLSE. As a result, new solutions are obtained in the form of solitons, kink–anti-kink type solitons, bright–dark solitons, traveling wave, trigonometric functions, elliptic functions and periodic solitary wave solutions. These new different types of solutions show the power and fruitfulness of this new method and also show two- and three-dimensional graphically with the help of computer software Mathematica. These new solutions have many applications in the field of physics and other branches of physical sciences. We can also solve other higher-order nonlinear partial differential equations (NPDEs) involved in mathematical physics and other various branches of physical sciences with this new technique.

Construction of bright–dark solitons and ion-acoustic solitary wave solutions of dynamical system of nonlinear wave propagation

The nonlinear (2 + 1)-dimensional Zakharov–Kuznetsov (ZK) equations deal with the nonlinear behavior of waves in collision-less plasma, which contains non-isothermal cold ions and electrons. Two-dimensional dust acoustic solitary waves (DASWs) in magnetized plasma, which consist of trapped electrons and ions are leading to (2 + 1)-dim (ZK) equation by using the perturbation technique. We found the solitary wave solutions of (2 + 1)-dimensional (ZK)-equation, generalized (ZK)-equation and generalized form of modified (ZK)-equation by implementing the modified mathematical method. As a result, we obtained the bright–dark solitons, traveling wave and solitary wave solutions. The physical structure of obtained solutions is represented in 2D and 3D, graphically with the help of Mathematica.

Construction of solitary wave solutions of some nonlinear dynamical system arising in nonlinear water wave models

The higher order of nonlinear partial differential equations in mathematical physics is studied. We used the analytical mathematical methods of the nonlinear (3+1)-dimensional extended Zakharov–Kuznetsov dynamical, modified KdV–Zakharov–Kuznetsov and generalized shallow water wave equations to demonstrate the efficiency and validity of the proposed powerful technique. The shallow water wave models have been applied in tidal waves and weather simulation. Exact wave solutions of these models in various forms such as Kink and anti-Kink solitons, bright–dark soliton, solitary wave and periodic solutions are constructed that have plenty of applications in diverse areas of physics. Graphically, we presented the movement of some obtained solitary wave solutions that aids in understanding the physical phenomena of these models.

Construction of soliton solutions of the modify unstable nonlinear Schrödinger dynamical equation in fiber optics

In this research article, we investigated the universal model of integrable system of modify unstable nonlinear Schrodinger equation. The mUNLSE described the disturbance of time period in slightly stable and unstable media and managed the instability of modulation wave train. We found the exact and solitary wave solutions of mUNLSE with the help of modified extended auxiliary equation mapping method. As a result, exact and solitary wave solutions in the form of elliptic functions, trigonometric functions, hyperbolic functions, bright and dark solitons, traveling wave, kink-type solitons and periodic solitary wave solution are obtained. These solutions show the power and effectiveness of this new method and two- and three-dimensional graphically with the help of computer software Mathematica. We can also solve other unstable nonlinear system of PDEs which are involved in Mathematical physics and many other branches of physical sciences with the help of this new method.

Solitary waves in a granular chain of elastic spheres: Multiple solitary solutions and their stabilities.

A granular chain of elastic spheres via Hertzian contact incorporates a classical nonlinear force model describing dynamical elastic solitary wave propagation. In this paper, the multiple solitary waves and their dynamic behaviors and stability in such a system are considered. An approximate KdV equation with the standard form is derived under the long-wavelength approximation and small deformation. The closed-form analytical single- and multiple-soliton solutions are obtained. The construction of the multiple-soliton solutions is analyzed by using the functional analysis. It is found that the multiple-soliton solution can be excited by the single-soliton solutions. This result is confirmed by the numerical analysis. Based on the soliton solutions of the KdV equation, the analytic solutions of multiple dark solitary waves are obtained from the original dynamic equation of the granular chain in the long-wavelength approximation. The stability of the single and multiple dark solitary wave solutions are numerically analyzed by using both split-step Fourier transform method and Runge-Kutta method. The results show that the single dark solitary wave solution is stable, and the multiple dark solitary waves are unstable.

Dipole and Combo Optical Solitons in Birefringent Fibers in the Presence of Four-Wave Mixing

The ultimate theme of this study is to develop dipole and combo optical solitons in birefringent fibers (BF) along with the effect of four-wave mixing (4WM). Two types of rough mediums are used which are Kerr law and parabolic law. Ansatz method of Choudhuri is applied to obtain dark in the bright (dipole) soliton solutions providing bright background for the propagation of optical dark pulse. Ansatz method of Li is applied to obtain combo soliton solutions providing bright solitary wave and dark solitary wave solutions. These results also exist to hold in non-Kerr media with higher order dispersion.

Modulation instability analysis of the generalized nonlinear Schrödinger equation and its bright, dark and complexiton soliton solutions

The generalized nonlinear Schrödinger (DNLS) equation depicts the propagation of light pulses in an optical fiber. We obtain a Lagrangian and the Hamiltonian system for the DNLS equation. Its bright, dark and bright–dark solitary wave solutions of the equation are derived by applying the amplitude ansatz method. In addition, we give some conditions to ensure the existence of solitons. The modulation instability analysis of the equation is also analyzed by using the standard linear stability analysis method. Finally, we derive the complexiton solutions of the DNLS equation. It is hoped that our results can be help to rich the dynamic behavior of the Schrödinger-type equations.

Matter-wave solitons of Bose–Einstein condensates with time-dependent complex potentials

To analytically investigate the matter-wave solitons of Bose–Einstein condensates (BECs) in time-dependent complex potential, we consider a cubic-quintic Gross–Pitaevskii (GP) equation with distributed coefficients and a dissipative term. By introducing a suitable ansatz, we establish the criterion of the modulational instability (MI) of the system and present an explicit expression for the growth rate of a purely growing MI. Effects of the parabolic background potential, as well as of the linear potential, the gain/loss parameter, and the two- and three-body interatomic interactions on the MI are investigated. We show how the feeding/loss parameter can be well used to control the instability of the system. The analytical resolution of the considered GP equation leads to exact bright, dark and kink solitary wave solutions which are used to investigate analytically the dynamics of matter-wave solitons in BECs under consideration. These analytical investigations show that the amplitude and the motion of bright, dark and kink solitary waves depend on the strengths of the two- and three-body interatomic interactions, as well as on the strengths of the external trapping potential and the parameter of the gain/loss of atoms in the condensate.

New exact solutions to the high dispersive cubic–quintic nonlinear Schrödinger equation

The complete discrimination system method is employed to find exact solutions for a dispersive cubic–quintic nonlinear Schrödinger equation with third order and fourth order time derivatives. As a result, we derive a range of solutions which include triangular function solutions, kink solitary wave solutions, dark solitary wave solutions, Jacobian elliptic function solutions, rational function solutions and implicit analytical solutions. Numerical simulations are presented to visualize the mechanism of Eq. (1) by selecting appropriate parameters of the solutions. The comparison between our results and other's works are also given.

Optical soliton and bright–dark solitary wave solutions of nonlinear complex Kundu–Eckhaus dynamical equation of the ultra-short femtosecond pulses in an optical fiber

In this research, we examine exact and solitary traveling solutions of nonlinear complex Kundu–Eckhaus equation with variable coefficients. Nonlinear complex Kundu–Eckhaus equation model has vital role in description of the proliferation of the pulses of the ultra-short femto second in an optical fiber, weakly nonlinear fragmented water waves, nonlinear optics and the quantum field theory. We implement a new auxiliary equation method and novel \(\left( \frac{G^{\prime }}{G}\right)\)-expansion technique on this model to obtain a new exact and solitary traveling wave solutions. The propose of this research is applying new method on this models to discover new form of soliton solutions that this makes the study of the electric and magnetic field of these pulses and the extent of their effect in the optical fiber more interesting.

Exact bright and dark solitary wave solutions of the generalized higher-order nonlinear Schrödinger equation describing the propagation of ultra-short pulse in optical fiber

Considering linear and nonlinear optical effects like group velocity dispersion, higher-order dispersion, Kerr nonlinearity, self-steepening, stimulated Raman scattering we obtain a higher-order nonlinear Schrödinger equation describing the propagation of ultra-short pulse in optical fiber. We construct exact bright and dark solitary wave solutions of the generalized obtained equation, obeying to some constraint relations between coefficient’s equation via the Bogning-Djeumen Tchaho-Kofané method (BDKm). The generalized higher-order nonlinear Schrödinger equation is obtained by affecting coefficients ni(i = 0, 1, .., 5) to different terms of non modified equation. New solutions are obtained, and the term or higher-order dispersion can be considered as the new selector of solitary wave-type propagating in the higher-order nonlinear optical fiber.

Solitons and Other Exact Solutions for Two Nonlinear PDEs in Mathematical Physics Using the Generalized Projective Riccati Equations Method

We apply the generalized projective Riccati equations method with the aid of Maple software to construct many new soliton and periodic solutions with parameters for two higher-order nonlinear partial differential equations (PDEs), namely, the nonlinear Schrödinger (NLS) equation with fourth-order dispersion and dual power law nonlinearity and the nonlinear quantum Zakharov-Kuznetsov (QZK) equation. The obtained exact solutions include kink and antikink solitons, bell (bright) and antibell (dark) solitary wave solutions, and periodic solutions. The given nonlinear PDEs have been derived and can be reduced to nonlinear ordinary differential equations (ODEs) using a simple transformation. A comparison of our new results with the well-known results is made. Also, we drew some graphs of the exact solutions using Maple. The given method in this article is straightforward and concise, and it can also be applied to other nonlinear PDEs in mathematical physics.

Bright–dark solitary wave and elliptic function solutions of unstable nonlinear Schrödinger equation and their applications

In this article, we study the unstable nonlinear Schrodinger equation (UNLSE). Analytically by modified extended direct algebraic method, which describes the disturbances in time evolution of marginally stable or unstable media. Explicit new exact solutions are derived in different form such as dark solitons, bright solitons, solitary wave, periodic solitary wave and elliptic function solutions of UNLSE. The movements of obtained solutions are shown graphically, which helps to understand the physical phenomenas of this unstable equation. Moreover, we also present the formation conditions of the bright and dark solitons of UNLSE. The obtained results and computational work shows the power and effectiveness of this method. Many other such types of nonlinear evolution equations arising in engineering, applied sciences and nonlinear optics can also be solved by this method.

Novel optical solitary waves and modulation instability analysis for the coupled nonlinear Schrödinger equation in monomode step-index optical fibers

This paper addresses the coupled nonlinear Schrödinger equation (CNLSE) in monomode step-index in optical fibers which describes the nonlinear modulations of two monochromatic waves, whose group velocities are almost equal. A class of dark, bright, dark-bright and dark-singular optical solitary wave solutions of the model are constructed using the complex envelope function ansatz. Singular solitary waves are also retrieved as bye products of the in integration scheme. This naturally lead to some constraint conditions placed on the solitary wave parameters which must hold for the solitary waves to exist. The modulation instability (MI) analysis of the model is studied based on the standard linear-stability analysis. Numerical simulation and physical interpretations of the obtained results are demonstrated. It is hoped that the results reported in this paper can enrich the nonlinear dynamical behaviors of the CNLSE.

Solitons in multi-body interactions for a fully modulated cubic–quintic Gross–Pitaevskii equation

• Dynamics of solitons in a fully modulated cubic–quintic Gross–Pitaevskii equation is analyzed. • Role of quintic parameter on the size of wave numbers of carrier and envelope is analyzed. • Lax-pair for the model with varying coefficients and external potential is given. • Regions of modulational instability and linear and nonlinear stabilities are investigated. A phase imprint approach is applied to a cubic Gross–Pitaevskii equation (GPE) in order to obtain a modified non-autonomous derivative cubic–quintic nonlinear GPE with fully variable coefficients. This model describes the dynamics of condensates in Bose–Einstein condensates, with both two- and three-body interatomic interactions with an external potential when the coefficient of the dispersion term is constant. This model is also applicable to fiber optics media in the absence of external potential. We show that this modified GPE model has a Lax-pair when all of its coefficients depend on time. However, the external potential depends on both the time and space variables. We obtain two classes of exact analytical solutions. These classes of solutions contain four different types of solitary-like wave solutions in the form of kink, anti-kink, bright, dark solitary, and periodic wave solutions. We reveal the effects of a quintic term on wave numbers, for both the carrier and envelope waves. Stability analysis is carried out, and conditions on parameters that determine regions with linear stability are discussed. Graphical analysis of some solutions are presented.

Modulation stability and optical soliton solutions of nonlinear Schrödinger equation with higher order dispersion and nonlinear terms and its applications

In optical fibers, the higher order non-linear Schrödinger equation (NLSE) with cubic quintic nonlinearity describes the propagation of extremely short pulses. We constructed bright and dark solitons, solitary wave and periodic solitary wave solutions of generalized higher order NLSE in cubic quintic non Kerr medium by applying proposed modified extended mapping method. These obtained solutions have key applications in physics and mathematics. Moreover, we have also presented the formation conditions on solitary wave parameters in which dark and bright solitons can exist for this media. We also gave graphically the movement of constructed solitary wave and soliton solutions, that helps to realize the physical phenomena's of this model. The stability of the model in normal dispersion and anomalous regime is discussed by using the modulation instability analysis, which confirms that all constructed solutions are exact and stable. Many other such types of models arising in applied sciences can also be solved by this reliable, powerful and effective method.