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In this paper, we consider the Sharma-Tasso-Olver-Like equation (STOLE) describing the dynamical behaviour of nonlinear dispersive waves in inhomogeneous medium. By using the Hirota bilinear method, some new type of solitons like breather-wave, kink solitary wave and rogue wave, one-, two- and new three-wave solutions to the STOLE have been determined. These results are achieved and verifed by using the Maple software. The obtained results are new from the existing results. For the further explanation of these solutions, diferent kinds of graphs are also drawn. These solitons suggest that this method is efective, straight forward and reliable as compare to other methods.

Some new optical solitons find for the Kudryashov equation (KE) in this study. These solutions are in the form of dark, bright, singular, singular-dark solitons and other solutions with certain conditions. These new solutions may be applied in the demonstration of Kudryashov equation in some better way. Modified integration method, the extended Sinh-Gordon equation expansion method (ShGEEM) is utilized to secure the aforesaid solutions. The results are also verified with the use of Mathematica symbolic programming software. These optical solitons suggest that this method is effective and straight forward compared to other methods and the results are reliable. The obtained solutions are very helpful in the field of optical fibers.

In this paper, we investigate the KdV equation with dual-power law nonlinearity. As a result, we have obtained general exact travelling wave soliton solutions such as bright soliton solution, dark soliton solution and periodic solution. These solutions have many free parameters in such away that they may be used to simulate many experimental situations. The main contribution in this work is to give the general solution of the obtained equations with different values of parameters $$n$$ .

In this paper, we consider the Sharma-Tasso-Olver-Like equation (STOLE) describing the dynamical behaviour of nonlinear dispersive waves in inhomogeneous medium. By using the Hirota bilinear method, some new type of solitons like breather-wave, kink solitary wave and rogue wave, one-, two- and new three-wave solutions to the STOLE have been determined. These results are achieved and verifed by using the Maple software. The obtained results are new from the existing results. For the further explanation of these solutions, diferent kinds of graphs are also drawn. These solitons suggest that this method is efective, straight forward and reliable as compare to other methods.

In this paper, conformable fractional resonant Schrodinger equation is studied. A new auxiliary equation approach is implemented to derive the solutions to the governing equation. Fractional complex traveling wave transform and homogeneous balance technique are the key procedures to implement the method. The predicted solution are set in finite series form of some functions satisfying an ODE of first order second degree. Many kinds of solutions covering exponential, hyperbolic and trigonometric functions are derived based on the suggested method.

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.

The modulational instability associated with discrete breathers in 2D quantum ultracold atoms is studied by using the Glauber’s coherent state combined with a semi-discrete approximation and multiple-scale methods. The linear stability analysis exhibits an intriguing threshold amplitude and instability regions associated with modulational growth rate. In addition, we demonstrate a coexistence of two bright intrinsic localized modes namely, the radial symmetric and bilateral symmetric modes, at the center and at the edges of the Brillouin zone, respectively, by alternating the on-site parameter interaction. Numerical investigations reveal a good agreement with the theoretical analysis.

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.

In this paper, we investigated the Gilson-Pickering (GP) equation and many new solutions are obtained with the aid of two different approaches, namely Jacobi elliptic functions and exponential rational function approach. Different choices of the parameters in obtained results lead to the solutions of some well known models, which are Camassa-Holm equation, the Fornberg-Whitham equation and the Rosenau-Hyman equation. The methods considered here can also help to have a panoply of new wave surfaces concerning other related partial differential equations. Further more, 2D and 3D graphical presentations of these surfaces are presented for the various parameters. Moreover, bifurcation behavior of nonlinear travelling waves of GP equation is discussed. Bifurcation theory of planer dynamical system is utilized to observe that considered model contains nonlinear periodic wave, bell shaped solitary wave and shock wave.

In this paper, using the continuum approximation and from a discrete equation governing a system of cold bosonic atoms in zig-zag optical lattices derived an equation in a continuum domain. With two different methods, namely the $$exp_a$$ function method and the Hyperbolic function method, found many solutions of the equation considered. These solutions are Hyperbolic solutions and trigonometric solutions. Discrete representations have been done to observe their behaviors. These results will help to understand the dynamics of cold bosonic atoms in zig-zag optical lattices and several other systems. More importantly, in this article derived the Lagrangian and the stability condition of the model studied. The methods employed here are very simple and concise.

The objective of this work is to study a monoatomic chain which is described by the Klein–Gordon model with first and second neighbors anharmonic interactions. We consider specifically the case where both of these interactions are of cubic–quartic type and show, through the rotating wave approximation, that envelop waves in our model are governed by a modified nonlinear Schrodinger equation. The modulational instability is investigated on the latter, with a particular attention on the impact of the second neighbors. It is then revealed that the second neighbors interaction increases the regions of instability in the space of parameters and, hence, expands the possibility of propagating solitary waves in the network. More importantly, we derive the exact solutions of our modified nonlinear Schrodinger equation and show the influence of the second neighbors on them. For example, the width of these solutions can be controlled by adjusting the parameters relative to the second neighbors. Numerical simulations are done in order to confirm the analytical studies. Equally, the methods used in this study can also be implemented on other nonlinear equations.

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.

Most of the important aspects of soliton propagation through optical fibers for transcontinental and transoceanic long distances can best be described using the nonlinear Schrödinger equation. Optical solitons are electromagnetic waves that span in nonlinear dispersive media and permit the stress and intensity to stay unaltered as a result of the delicate balance between dispersion and nonlinearity effects. However, this study exploited the Jacobi elliptic method and obtained different soliton solutions of the decoupled nonlinear Schrödinger equation with ease. Discussions about the obtained solutions were made with the aid of some 3D graphs.

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.

In this paper, a new exact solution of the conformable Gilson–Pickering equation is investigated. It should be noted that some of the individual cases of the Gilson–Pickering equation are the conformable Camassa–Holm, the conformable Fornberg–Whitham, and the conformable Rosenau–Hyman equations. A new version of modified Kudryashov’s method with the help of the mathematical software package is employed to carry out this aim. It is believed that the new modified Kudryashov’s method is well suited, such that it can adapt to a broad range of partial differential equations.

In this paper, we study a nonlinear Schrodinger equation with quintic nonlinearity, self-steepening and self-frequency shift terms describing the polarization mode in an optical fiber. As results, several new chirped soliton solutions not yet reported in the literature are obtained. These solutions are found without using computer codes. The solutions are bright, dark and cnoidal solitons. The method used here is very effective and simple and can be applied to other types of nonlinear equations.

In this research, we work on a specific class of nonlinear evolution equation which is the modify unstable nonlinear Schrodinger equation. This equation is used to describe a time evolution of disturbances in unstable media. Various solutions have been obtained. The results deduced are of varied types and include bright solution, dark solution, rational dark-bright solution, as well as cnoidal solutions. These solutions might be useful in engineering fields. Some conditions for the stability of these solutions are presented. The method used here is understandable and very powerful for solving the nonlinear problems.

In this work, we use the new Jacobi elliptic function expansion method to find exact soliton solutions for a discrete nonlinear electrical transmission line in (2 + 1) dimension. Several new solutions have been obtained. The solutions found by the current method are of varied types and include hyperbolic and trigonometric solutions, as well as Jacobi elliptic solutions. We show that the existence of these solutions depends on the parameters of the network. Comparisons of our new results with the well-known results are obtained. The solutions found here may be also used in optical fibers to transport information. The method applied here is very simple and concise and can be also applied to other nonlinear partial differential equations.

In this paper, we apply two methods which are the arbitrary nonlinear parameters and the exponential rational function method to construct many new exact solutions of the higher-order nonlinear partial differential equations, namely, the higher-order nonlinear Schrodinger (HNLS) equation. The solutions obtained by the current methods are generalized periodic solutions. The shape of the solutions can be well controlled by adjusting the parameters of the system. Optical soliton solutions obtained can be used to transport information in the telecommunication domain. It also comes from this work that the behavior of this HNLS equation may be easily studied by means of the phase plane plot which is the best tool to predict some solutions.

In this work, the dynamics of modulated waves in a modified Noguchi nonlinear electrical transmission line is studied with the contribution of second neighbors. It comes from this analysis that the line is governed by a dissipative nonlinear Schrodinger equation. One observes that the second neighbors counterbalance the effect of the linear capacitor CS in the frequency domains. The second neighbors well influence the line by increasing its bandwidth, its group velocity and the magnitude of the wave during its propagation. In the dispersion curve, we show that there exits a new region for the modulational instability/stability compared to the work of Pelap et al. (Phys. Rev. E 91, 022925 (2015)). The exactness of the analytical studies is accredited by numerical calculations. The most important feature of the new region, i.e. the second neighbors, is that the same frequency allows the use of either a bright soliton or a dark soliton depending on the choice of an appropriated wavelength.

In this article, we employ the perturbed Fokas-Lenells equation (FLE), which represents recent electronic communications. The Riccati-Bernoulli Sub-ODE method which does not depend on the balance rule is used for the first time to obtain the new exact and solitary wave solutions of this equation. This technique is direct, effective and reduces the large volume of calculations.

This work investigates the dynamics of modulated waves in a coupled nonlinear LC transmission line. By means of a method based on the semi-discrete limit and in suitably scaled coordinates, we derive the two-dimensional NLS equation governing the propagation of slowly modulated waves in the network. The exact transverse solution is found and the analytical criteria of stability of this solution are derived. The condition for which the network can exhibit modulational instability is also determined. The exactness of this analytical analysis is confirmed by numerical simulations performed on the exact equation of the network.

We report in this work, an analytical study of quantum soliton in 1D Heisenberg spin chains with Dzyaloshinsky-Moriya Interaction (DMI) and Next-Nearest-Neighbor Interactions (NNNI). By means of the time-dependent Hartree approximation and the semi-discrete multiple-scale method, the equation of motion for the single-boson wave function is reduced to the nonlinear Schrodinger equation. It comes from this present study that the spectrum of the frequencies increases, its periodicity changes, in the presence of NNNI. The antisymmetric feature of the DMI was probed from the dispersion curve while changing the sign of the parameter controlling it. Five regions were identified in the dispersion spectrum, when the NNNI are taken into account instead of three as in the opposite case. In each of these regions, the quantum model can exhibit quantum stationary localized and stable bright or dark soliton solutions. In each region, we could set up quantum localized n-boson Hartree states as well as the analytical expression of their energy level, respectively. The accuracy of the analytical studies is confirmed by the excellent agreement with the numerical calculations, and it certifies the stability of the stationary quantum localized solitons solutions exhibited in each region. In addition, we found that the intensity of the localization of quantum localized n-boson Hartree states increases when the NNNI are considered. We also realized that the intensity of Hartree n-boson states corresponding to quantum discrete soliton states depend on the wave vector.

The resonant nonlinear Schrödinger equation is studied in this work with the aid of two methods, namely the exponential rational function method and the modified exponential function method. This equation is used to describe the propagation of optical pulses in nonlinear optical fibers. Being concise and straightforward, these methods are used to build new exact analytical solutions of the model. The solutions obtained are not yet reported in the literature. The methods proposed can be extended to other types of nonlinear evolution equations in mathematical physics.