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This paper investigates the pure-cubic complex Ginzburg–Landau equation (PC-CGLE) with different nonlinearities such as Kerr law, power law and so on. We get the dynamic systems and show that solitons and periodic solutions exist through the complete discrimination system for the polynomial method (CDSPM). To verify these conclusions, we construct the traveling wave solution via the CDSPM, and some new solutions are also built. The soliton stability and modulation instability with two types of nonlinearities are discussed. Finally, by adding perturbed terms to the dynamic system, we obtain the largest Lyapunov exponents and the phase diagrams of the equation, which proves there are chaotic behaviors in PC-CGLE. The results such as Gaussian soliton solutions and chaotic behavior for PC-CGLE are initially discovered in the present paper. • Investigate the pure-cubic complex Ginzburg–Landau equation (PC-CGLE) with various nonlinearities, especially the power law and the triple-power law. • Demonstrate the existence of solitons and periodic solutions through the application of the complete discrimination system for the polynomial method. • Construct traveling wave solutions to obtain all the exact solutions and construct some new solutions, thus enriching the conclusions. • Soliton solutions' stability and the equation's modulation instability are analyzed and demonstrated. Chaotic behaviors in PC-CGLE are demonstrated by introducing perturbed terms, marking the initial identification of this phenomenon. • Some results like Gaussian soliton solutions and chaotic behaviors to PC-CGLE are initially given. [ABSTRACT FROM AUTHOR]

In this paper, we consider the linear stability of traveling wave solutions for one-dimensional compressible isentropic Navier-Stokes equations with Maxwell's Law. The global stability of traveling wave solution is established with shock-profile initial data for the linearized system. Anti-derivative and some delicate energy methods are explored to get the desired results. Moreover, the relaxation limit of traveling wave solution is also obtained. [ABSTRACT FROM AUTHOR]

This paper considers the generalized Davey-Stewartson equation that is used to investigate the dynamics of wave propagation in water of finite depth under the effects of gravity force and surface tension. The model is considered in the presence of full nonlinearity. The main objective of this paper is to extract soliton solutions of the generalized Davey-Stewartson equation. Three state-of-the-art integration schemes, namely exp (- Φ (ξ)) -expansion method, the first integral method and the Sine-Gordon expansion method have been employed for obtaining the desired soliton solutions. The proposed methods successfully attain different structures of explicit solutions such as bright, dark, singular, rational and periodic solitary wave solutions. All the newly found solutions are discussed with their existence criteria. The 2D and 3D portraits are also shown for some of the reported solutions. [ABSTRACT FROM AUTHOR]

In this work, (1 + 2)-dimensional chiral nonlinear Schrödinger's equation is considered. The optical soliton solutions for the aforementioned model are extracted in a systematic way. Two robust and efficient integration tools such as the first integral method and the sine-Gordon expansion method are used to extract abundant solution such as dark, bright soliton as well as complexitons. The parametric conditions for the validity of these solutions are also listed along with the solutions. The physical meaning of the geometrical structures for some solutions is discussed for different choices of the free parameters. [ABSTRACT FROM AUTHOR]

The main object of this paper is to study the bifurcation, chaotic pattern and traveling wave solution of the perturbed stochastic nonlinear Schrödinger equation with generalized anti-cubic law nonlinearity and spatio-temporal dispersion. A traveling wave transformation is used to simplified the perturbed stochastic nonlinear Schrödinger equation into ordinary differential equation. The dynamic behavior of two-dimensional planar dynamical systems and their perturbed systems are studied, and bifurcation, phase portrait, and Poincaré section are presented. Furthermore, traveling wave solutions included Jacobian function solutions, trigonometric function solutions and hyperbolic function solutions are constructed. • The main purpose of this paper is to study the chaotic pattern and traveling wave solution of the perturbed stochastic nonlinear Schrödinger equation with generalized anti-cubic law nonlinearity and spatio-temporal dispersion. • The dynamic behavior of two-dimensional planar dynamical system and their perturbed system are studied by using the theory of dynamics system. • Traveling wave solutions included Jacobian function solutions trigonometric function solutions and hyperbolic function solutions are constructed. [ABSTRACT FROM AUTHOR]

This article investigates the traveling wave solution of the parabolic law nonlinear stochastic dispersive Schrödinger-Hirota equation with multiplicative white noise, and its conclusions can be applied to optical communication. Firstly, the equation is transformed into a nonlinear ordinary differential equation through traveling wave transformation, and its planar dynamic system is studied. The phase portraits of the equation in various cases are drawn by Maple software. Secondly, the traveling wave solutions of this equation are obtained by using the complete discriminant system theory of cubic polynomials. Finally, we draw two-dimensional graphics, three-dimensional graphics, density plots and contour plots for the module length of some solutions under specific parameters. In addition, the above research results can help readers have a more comprehensive and clear understanding of this equation and its application. • The paper studies the traveling wave solution and dynamics analysis of the parabolic law nonlinear stochastic dispersive Schrödinger-Hirota equation with multiplicative white noise. • The planar dynamics system of the Schrödinger-Hirota equation is discussed. • Three-dimension graphic, two-dimensional graphic, density plot and contour plot of some obtained solutions are plotted. [ABSTRACT FROM AUTHOR]

In this paper, the traveling wave solutions of the model of magneto-optic waveguide with anti-cubic nonlinearity are obtained by using the complete discrimination system for polynomial method, including singular solutions, solitary wave solutions,and double periodic solutions. And under specific parameter conditions, three types of optical wave patterns are obtained to visualize the model and demonstrate their accurate physical behavior. Then the dynamic properties of Schrödinger equation in magneto-optic waveguide with anti-cubic nonlinearity are analyzed, the existence of periodic and solitary solutions is proved based on the bifurcation method. Also the Hamiltonian properties and the classification of its equilibrium points are obtained. In final, we analyze the chaotic behavior of the model under some external perturbations. • This paper use the traveling wave transformation to convert the equation into a dynamic system and obtain its Hamiltonian properties. • Based on the bifurcation method, the existence of cycles and solitary solutions is proved. • The equation is analyzed using the complete discrimination system for polynomial method, and all its traveling wave solutions are obtained. • The external perturbation terms, in order to analyze the chaotic behavior of this equation. [ABSTRACT FROM AUTHOR]

In this article, we investigate the exact traveling wave solutions of two nonlinear time fractional wave equations. The fractional derivatives are described in the sense of conformable fractional derivatives. In addition, the traveling wave solutions are accomplished in the form of hyperbolic, trigonometric, and rational functions involving free parameters. To investigate such types of solutions, we implement the new generalized ( G ′ / G ) -expansion method. The extracted solutions are reliable, useful and suitable to comprehend the optimal control problems, chaotic vibrations, global and local bifurcations and resonances, furthermore, fission and fusion phenomena occur in solitons, the relativistic energy-momentum relation, scalar electrodynamics, quantum relativistic one-particle theory, electromagnetic interactions etc. The results reveal that the method is very fruitful and convenient for exploring nonlinear differential equations of fractional order treated in theoretical physics. [ABSTRACT FROM AUTHOR]

This article mainly studies the new traveling wave solutions of the stochastic modified Korteweg–de Vries equation with multiplicative noise. The traveling wave solutions in the form of hyperbolic function, trigonometric function, rational function and Jacobi elliptic function are obtained by transforming the modified Korteweg–de Vries equation into an ordinary differential equation through traveling wave transformation, and combining it with the complete discriminant system of polynomials. This article fully applies visual analysis technology and provides graphical representations of the obtained solutions in multiple dimensions such as phase maps, sensitivity maps, 3D maps and 2D maps, providing an intuitive and convenient channel for people to further understand the physical characteristics of the solutions for the stochastic modified Korteweg–de Vries equation. This article provides a rich graphical analysis of the solutions to the stochastic modified Korteweg–de Vries equation from multiple perspectives, and provides a fast and efficient solution method for the exact solution of the equation. The research method in this paper can also be used to study nonlinear partial differential equation in other fields. • The paper studies the traveling wave solutions of the stochastic modified Korteweg–de Vries equation. • This article provides a rich graphical analysis of the solution of the equation. • The solution process is simple and fast, and the solution method is effective. [ABSTRACT FROM AUTHOR]

This article investigates the traveling wave solutions of the stochastic Davey–Stewartson equations based on the theory of the complete discriminant system of quadratic polynomials. Firstly, the stochastic Davey–Stewartson equations are transformed into the nonlinear ordinary differential equations through traveling wave transformation. By using Maple software, some two-dimensional and three-dimensional phase portraits are drawn. Secondly, according to the classification of the roots of quadratic polynomial equation, new traveling wave solutions of the stochastic Davey–Stewartson equations are obtained by using the complete discriminant system, which include rational function solutions, triangle function solutions and the Jacobi elliptic function solutions. Finally, three-dimensional-surface plots and two-dimensional-shape plots of some obtained solutions are drawn by using Maple software. • The paper studies multiplicative brownian motion stabilizes traveling wave solutions of the stochastic Davey–Stewartson equations. • The bifurcation of the dynamics system of the stochastic Davey–Stewartson equations are discussed. • Three-dimension graphic, two-dimensional graphic of some obtained solutions are plotted. [ABSTRACT FROM AUTHOR]

In this paper, the stochastic space-fractional long-short-wave interaction system (SF-LSWIS) with multiplicative white noise is considered. The stochastic exact solutions, including triangular function solutions, hyperbolic function solutions, and Jacobi elliptic function solutions, are obtained via the complete discriminant system for polynomial method. For this purpose, the traveling wave transformation is applied to carry out the ordinary differential equation of this model, which is also converted to a dynamical system. Thereafter, qualitative behavior and bifurcation of the phase portraits of the dynamical system are investigated using the qualitative theory of dynamical systems. The chaotic behaviors of the system considering external periodic perturbation are studied. Furthermore, the 3D surface, graphs of contour plots, and level curves of some exact solutions are depicted by choosing proper parameter values. The influence and effect of fractional derivatives and noise strength on the solutions are also studied. • The stochastic space-fractional long-short-wave interaction system is studied. • The exact solutions to this model are obtained using the complete discrimination system for polynomial method. • Phase portrait, bifurcation, and chaotic pattern of the equation are studied. [ABSTRACT FROM AUTHOR]

The paper mainly studies the traveling wave solutions, phase portrait and chaotic pattern of the generalized fractional stochastic Schrödinger equations forced by multiplicative Brownian motion. By using the polynomial complete discrimination method, the hyperbolic functions, rational functions and Jacobi elliptic function solutions are obtained. Moreover, the physics properties of the obtained solutions are shown through three-dimensional, two-dimensional and contour graphs. In order to help better analyze dynamic behavior, phase diagrams, chaos behavior and sensitivity are plotted by using Maple software. • The paper studies the traveling wave solutions of the generalized fractional stochastic Schrödinger equations. • Phase portrait and chaotic pattern are presented. • The solution process is simple and fast, and the solution method is effective. [ABSTRACT FROM AUTHOR]

In this paper, the coupled Kundu–Mukherjee–Naskar equation is considered, which is usually used to describe the oceanic rogue waves, hole waves and Bragg grating fibers. Firstly, the traveling wave transformation is applied to convert the coupled Kundu–Mukherjee–Naskar equation into two-dimensional planar dynamic system. Secondly, bifurcation and two-dimensional phase portrait of the coupled Kundu–Mukherjee–Naskar equation are discussed by using the plane dynamics system. What is more, after adding periodic perturbation term, two-dimensional phase portrait, three-dimensional phase portrait, Poincaré section and sensitivity of its perturbed system are plotted by the Maple software. Finally, the optical soliton solutions of the coupled Kundu–Mukherjee–Naskar equation are obtained by using planar dynamical system method. • The main purpose of this paper is to study the chaotic pattern, bifurcation, sensitivity and traveling wave solutions of the coupled Kundu–Mukherjee–Naskar equation. • The bifurcation of the dynamics system of the coupled Kundu–Mukherjee–Naskar equation are discussed. • Two-dimensional phase portraits, three-dimensional phase portraits, Poincaré sections and sensitivity analysis of the dynamics system with perturbation term are drawn. [ABSTRACT FROM AUTHOR]

This study employed the planar dynamics system method to investigate the chaotic behavior and traveling wave solution of the fractional stochastic Zakharov system arising in plasma physics. Through the analysis of the phase portraits, the sensitivity of the dynamic system has been examined and various new exact solutions for traveling waves have been constructed. These obtained solutions are conducive to further understanding the propagation of plasma waves. Furthermore, an analysis of the impact of fractional derivative and random noise on the solutions is done. The chaotic behavior of dynamic systems is examined by time series, Poincaŕe sections, and 2D and 3D phase portraits. Finally, the comparative investigation reveals that the characteristics of the solutions are significantly impacted by both the fractional derivative and random noise. The research methodologies and results are efficacious and can be extended to the exploration of more complex phenomena. • The chaotic behavior of the fractional stochastic Zakharov system is studied, and the exact solution is constructed. • The governing equation is set as a nonlinear partial differential equation in the sense of Stratonovich random integral and conformable fractional derivative. • The comparative analysis results show the effectiveness of the influence of fractional derivative and noise intensity on the system solution. [ABSTRACT FROM AUTHOR]

Analytical solutions are the most effective tools for understanding phenomena modeled with differential equations. On the other hand, analytical solvers may be limited in their ability to handle complex problems and we are forced to rely on approximate and numerical methods. Through the investigation of two types of fractional operators, this article aims to investigate both approaches to studying the interactive dynamics of an eco-epidemiological diffusion model. We consider two differential operators in our main model: local beta time and non-local Riesz space fractional derivative. Firstly, we propose novel classes of soliton wave solutions for the local time fractional version of the system using two efficient analytical methods. In the second part of the paper, we consider the case where the second-order space derivative operator in the problem is of the type of Riesz fractional operator that is non-local. A finite difference scheme is the main idea behind discretizing this fractional system. Throughout this article, we highlight the validity and efficiency of both analytical and numerical perspectives. The results can be examined by epidemiologists for a more comprehensive interpretation. • Two fractional operators are considered in an eco-epidemiological model. • Exact solutions to the model with local β -derivative are found via two analytical techniques. • We also investigate approximate solutions to the model with a nonlocal fractional operator. • Numerical simulations corresponding to the obtained results are provided. • The acquired results are new and have not been obtained for the model in previous papers. [ABSTRACT FROM AUTHOR]

This research aims to look into alternative unique solitary wave solutions to the perturbed Chen–Lee–Liu (CLL) equation to explain the kinetic and physical aspects of an optical fiber pulse. The perturbed CLL equation is one of the most well-known icon models derived from the well-known Schrödinger equation. Two different analytical methodologies are used to develop novel solitary wave solutions. These responses are then thoroughly evaluated using a well-known numerical approach to establish their underlying veracity. Various graphics are used to show the pulse wave analysis process and outcomes in an optical cable. We show how scientifically unique the study is by explaining the comparison figure that was made from our data and has been used in many recent research studies • Study of perturbed Chen–Lee–Liu equation. • Analytical and numerical novel constructed results. • Investigating the obtained results' accuracy. • Explaining the obtained solutions through some distinct types of sketches. [ABSTRACT FROM AUTHOR]

In this work, the modified simple equation (MSE) method is applied to some class of nonlinear PDEs, namely, a system of nonlinear PDEs, a (2 + 1)-dimensional nonlinear model generated by the Jaulent–Miodek hierarchy, and a generalized KdV equation with two power nonlinearities. As a result, exact traveling wave solutions involving parameters have been obtained for the considered nonlinear equations in a concise manner. When these parameters are chosen as special values, the solitary wave solutions are derived. It is shown that the proposed technique provides a more powerful mathematical tool for constructing exact solutions for a broad variety of nonlinear PDEs in mathematical physics. [ABSTRACT FROM AUTHOR]

In the paper, the nonlinear variable-coefficient fifth-order Schrödinger (NLVS) equation is researched. The NLVS equation is an integrable equation, which can be described the spreading of ultrashort pulses in an inhomogeneous optical fiber. Firstly, by using the modified traveling wave transformation, the NLVS equation is changed into an ordinary equation. Secondly, by the Jacobian elliptic function expansion method for the ordinary equation, we obtain the exact solutions for the ordinary equation, and then, we obtain the exact solutions to the NLVS equation. These solutions mainly include three types: Jacobi elliptic function solutions, hyperbolic function solutions, and triangular function solutions. Finally, according to the special parameters, we show the figures of the exact solutions. • By the modified traveling wave transformation method, we obtain new explicit solutions for the nonlinear variable-coefficient fifth-order Schrödinger equation. • There exist mainly trigonometric function, hyperbolic function, and rational function traveling wave solutions. • Physically, we explain all the solutions, which include compacton, soliton, cuspon, and peakon traveling waves. [ABSTRACT FROM AUTHOR]

In this work, the exact traveling wave solutions of the generalized Fisher's equation and modified equal width equation are studied by using the (G/G)-expansion method. As a result, many solitary wave solutions are derived from the solutions via hyperbolic functions, trigonometric functions and rational functions. When the parameters were taken at special values, the results obtained were compared with the solution via the tanh method established earlier. In fact, many general nontraveling wave solutions are obtained. The efficiency of the method is demonstrated by applying it for a variety of selected equations. [ABSTRACT FROM AUTHOR]

In this article, the dynamics of modified nonlinear waves for the generalized breaking soliton (gBS) equation have been explored by using analytical and numerical approaches. Firstly, the Tanh method is used to derive the solitonic solutions of the considered equation and their graphical illustrations for different values of parameters have been discussed. By using the proposed strategy, a variety of solutions can be found. Furthermore, the Galilean transformation is used to investigate the bifurcation behavior of the model. All possible phase portraits have been drawn and examined in detail. More specifically, it is shown that under defined parametric restrictions, the model has a straightforward specific value. At each fixed point, the existence of periodic points and bifurcation analysis is also confirmed. To clarify theoretical results, detailed numerical simulations are offered. Then by introducing perturbed term, the periodic and quasi-periodic behaviors of the discussed equation for various initial values are reported. Different periodic solutions can also be found in this parameter zone. Furthermore, it is reported that discussed equation is multi-stable for different values of parameters. Finally, sensitivity analysis for different initial value problems has been elaborated. • Solitonic structures of generalized breaking soliton. • Transformation of the considered model in the planer dynamical system. • Investigate the bifurcation behavior of the model by Galilean transformation. • Periodic and quasi-periodic behaviors of the (gBS) equation. • Multistability, Dynamic behavior of non-linear wave solutions of perturbed term with (gBS) equation. [ABSTRACT FROM AUTHOR]

Abstract: The homotopy perturbation method (HPM) is employed to find the explicit and numerical traveling wave solutions of Whitham–Broer–Kaup (WBK) equations, which contain blow-up solutions and periodic solutions. The numerical solutions are calculated in the form of convergence power series with easily computable components. The homotopy perturbation method performs extremely well in terms of accuracy, efficiently, simplicity, stability and reliability. [Copyright &y& Elsevier]

In the paper, we study the Boiti-Leon-Manna-Pempinelli equation with (3 + 1) dimension. By using the modified hyperbolic tangent function method, we obtain more new exact solutions for the (3 + 1) -dimensional Boiti-Leon-Manna-Pempinelli equation, which expand the solutions in previous literature. Those solutions have three forms determined according to the choice of parameters: trigonometric function form, rational function form, hyperbolic function form. Both trigonometric function form solutions and hyperbolic function form solutions include logarithmic function. And we give some examples and draw the some figures of the exact solutions. [ABSTRACT FROM AUTHOR]

The traveling wave solution of generalized hyperelastic-rod wave equation is studied and necessary conditions to the existence of lubricous traveling wave solution are discussed. This paper studies the specific mode and the existence of traveling wave solution to the equation, in which g(u) is a cubic polynomial function or an exponential function. Via phase space (φ, η) , the equilibrium point and trajectory of this equation are discussed. [ABSTRACT FROM AUTHOR]

We analyze the paper "Construction of traveling waves patterns of (1 + n)-dimensional modified Zakharov-Kuznetsov equation in plasma physics" by Adil Jhangeer and et al. [RINP, 2020, 103330] and demonstrate that results of this paper are wrong. The correct results are given. [ABSTRACT FROM AUTHOR]

Abstract: The recently published paper “Jacobi elliptic function solutions for the modified Korteweg–de Vries equation” [J. King Saud Univ. Sci. 25 (2013) 271–274] is analyzed. We show that these Jacobi elliptic function solutions obtained by the authors do not satisfy the original modified Korteweg–de Vries equation. [Copyright &y& Elsevier]