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In this work, we propose a new model concerning emphatically coupled nerve fibers. Through the fractional complex transformation and the improved Riccati equation mapping method, various exact traveling wave solutions for the considered model are derived. In our resolution, we introduce the non-integer derivative for the description of long memory and hereditary properties of various materials. These phenomena have not yet been taken in account in the previous works. This memory effect can be used to describe some unknown phenomena observed during the propagation of impulses in myelinated fibers.

In this work, the energy spectrum and intrinsic localized modes associated with the modulational instability of boson chains is described by a Bose–Hubbard model. By using of number states method combined with the numerical diagonalization, we show that the energy spectrum of the system exhibits bound states signature. With the help of multiple scales method in addition to a quasidiscreteness approximation, we obtain bright and dark type intrinsic localized modes at the center and at the edges of the Brillouin zone respectively and their appearance conditions. We found that the inclusion of interaction of two bosons at the same site can influence the appearance conditions of bright and dark soliton solutions. Considering the fact that nowadays, the forthright way to predict the formation of intrinsic localized modes in nonlinear systems is the modulational instability. We investigate analytically, through the linear stability of plane wave solutions, the existence of localized structures in the Bose–Hubbard chain. It is found that the shape of the region of modulational instability and the instability growth rates of the plane wave can be more affected dramatically when the interaction of two bosons at the same site is involved in the system. Furthermore, we calculate the instability growth rates of plane wave at the center and at the edge of the Brillouin zone for different values of the interaction of two bosons to analyze their formation conditions which are in full agreement with the method of multiple scale in addition to a quasi-discreteness approximation analysis.

This work investigates the ability of a new tuned pendulum mass damper to control the behavior of a structure. A very rich nonlinear dynamics is reached by a mass fixed on a spring and driven by a conveyor belt. That is the stick-slip motion (either regular or chaotic), the chaotic motion and very low amplitude motion. When a control systems like the tuned mass damper are used, the conversion of chaotic stick-slip into regular stick-slip is observed. For certain tuned mass dampers, the regular stick-slip is transformed into a stick-slip with very small amplitude. In addition, the chaotic motion is converted into regular motion, while very low amplitude motion remains unchanged. Very few cases of anti-control have been noted. Thereafter, a comparison on the effectiveness of control systems was made, especially in the case of chaotic motions. For this purpose, the largest Lyapunov exponent is evaluated and the stability map was obtained. It is shown that the best control system depends on the value of the mass ratio between the controller and the structure (which should be as small as possible for civil engineering implementation). In addition, the best control system is closely related to the type of motion considered.

This paper presents new exact analytical solutions of a nonlinear Schrodinger equation with a cubic–quintic nonlinearity and in presence of a couple of perturbation terms. This equation describes the dynamics of soliton propagation through an optical fiber. Several solutions are found without applying the computer codes and by considering the integration constant. The solutions are bright, dark and cnoidal solitons. These solutions may have significant applications in telecommunication systems where solitons are used to codify or for the transmission of data. The method used here is very effective and powerful and can be applied to other types of nonlinear equations.

The dynamics and modulation instability of an ultracold gas of bosonic atoms in an optical lattice can be portrayed by a Bose–Hubbard model and the system parameters are mastered by laser light. Based on the time dependent Hartree approximation combined with the semi-discrete multiple-scale method, the equation of motion for single-boson wave function is found analytically, the existence conditions of appearance of bright stationary localized solitons solutions of this quantum Bose–Hubbard model are discussed. We find that the introduction of the next nearest neighbor interactions (NNNI) may change the stability property of the plane waves and may predict the formation of modulational instability in the wave number k = kmax and k = keBZ in the system. With the help of stationary localized single-boson wave functions obtained, the quantized energy level and the quantum breather state are determined. The performance of the analytical results are checked by numerical calculations. Furthermore, we have shown that the presence of the NNNI affect significatively the shape of the region of modulational instability and it is responsible of the appearance of new region of modulational instability that occurs for the k = kmax carrier wave. The formation conditions of the modulational instability region predicted by the analytical analysis of stationary localized solutions in the wave number k = kmax and k = keBZ are in good agreement with the forecast respectively.

This paper presents many new solutions of a modified Zakharov–Kuznetsov equation obtained by using the Jacobi elliptical function method. This equation is shown to model a two dimensional discrete electrical lattice. The solutions reported herein are of varied types and include hyperbolic and trigonometric solutions, as well as kink and bell-shaped solitons. The comparison of our results to well-known ones is done. The method used here is very simple and concise and can be also applied to other nonlinear partial differential equations. More importantly, the solutions found in this work can have significant applications in telecommunication systems where solitons are used to codify data.

In order to investigate the nonlinear fractional Zoomeron equation, we propose three methods, namely the Jacobi elliptic function rational expansion method, the exponential rational function method and the new Jacobi elliptic function expansion method. Many kinds of solutions are obtained and the existence of these solutions is determined. For some parameters, these solutions can degenerate to the envelope shock wave solutions and the envelope solitary wave solutions. A comparison of our new results with the well-known results is made. The methods used here can also be applicable to other nonlinear partial differential equations. The fractional derivatives in this work are described in the modified Riemann–Liouville sense.

The aim of this paper is to apply the exponential rational function method for solving nonlinear equations arising in various physical models such as nonlinear electrical transmission lines, optical fibers, DNA, to mention a few. New exact solutions are obtained and can help to well understand the process of those systems. The method used is very effective and simple and can be applied to other types of nonlinear equations.

In this paper, by using the new Jacobi elliptic function rational expansion method and the exponential rational function method, new exact solutions of the unstable nonlinear Schrodinger equation are obtained. The present methods are very effective and simples and can be applied to other types of nonlinear equations. The results found can be compared with other methods used to solve the same model.

We probe in detail, properties of solitons and modulational instability (MI) in addition to the generation of soliton-like excitations in a discrete two-dimensional (2D) Ablowitz-Ladik (AL) equation. Using the multiple-scale and Rayleigh-Ritz variational methods, we get bright and dark radial solitons. As bright radial soliton is linked to MI, we have predicted the appearance of instability areas for the wave number belonging to [ 0 , π / 2 [ in the Brillouin zone. Through the linear stability analysis, the MI criterion of appearance of instability areas are found. Furthermore, we show that the growth rate of the amplitude of MI may be significantly influenced by the attractive nonlinearity term. Numerical simulations are performed to support our analytical analysis and an excellent agreement has been obtained.

This paper studies the (2 + 1)-dimensional hyperbolic nonlinear Schrodinger equation. The first integral of this equation, the phase portraits and the effective potentials are provided. Two different methods are applied to find exact analytical solutions. These methods are the arbitrary nonlinear parameters and the new Jacobi elliptic function expansion method. To give a behavior of the equation studied, some representations are done. In the context of mono-mode optical fibers and in many other domains like nonlinear transmission lines, Bose-Einstein capacitors and so on, the results obtained may be used. We have also established that the solutions obtained here are different from those encounter in the literature concerning the same model.