Traveling Wave Reduction Research Articles

Conservation laws of a generalized model for propagation pulses with four power nonlinearities

A generalized model for propagation pulses with four power nonlinearities is considered. The equation studied is the generalization of some well-known models and allows us to evaluate the influence of various processes on pulse propagation. The three conservation laws of the equation are found. The equation does pass not the Painlevé test and the Cauchy problem cannot be solved by the inverse scattering transform. Analytical solutions of the generalized nonlinear Schrödinger equation are found taking into account the traveling wave reduction. Optical solitons corresponding to the mathematical model are given. Conservative quantities for the bright optical soliton are calculated.

Painlevé Analysis of the Traveling Wave Reduction of the Third-Order Derivative Nonlinear Schrödinger Equation

The second partial differential equation from the Kaup–Newell hierarchy is considered. This equation can be employed to model pulse propagation in optical fiber, wave propagation in plasma, or high waves in the deep ocean. The integrability of the explored equation in traveling wave variables is investigated using the Painlevé test. Periodic and solitary wave solutions of the studied equation are presented. The investigated equation belongs to the class of generalized nonlinear Schrödinger equations and may be used for the description of optical solitons in a nonlinear medium.

Rogue periodic waves of the short pulse equation and the coupled integrable dispersionless equation

In this paper, we explore the topic of rogue waves on periodic backgrounds. Using the travelling wave reduction method, we derive two families of periodic solutions for the coupled integrable dispersionless (CD) equation expressed in Jacobi elliptic dnoidal and conoidal functions, respectively. We use these Jacobi elliptic functions as seed solutions to construct algebraic decaying solitons on the dnoidal background of the CD equation based on the one-fold Darboux transformation (DT). Furthermore, we utilize the two-fold DT to find rogue wave solutions on the conoidal background of the CD equation. To obtain solutions of the short pulse equation, we use the hodograph transformation. We analyse the dynamics of the explicit solutions obtained from this research through plots.

Analytical solutions of the generalized Kaup–Newell equation

The generalized Kaup–Newell equation for describing the pulse propagation in a nonlinear optical medium is considered. The traveling wave reduction of this equation is taken into account. The method of direct transformations is used to construct a general solution of a nonlinear ordinary differential equation. Analytical solutions are found, expressed in terms of elliptic integrals in canonical form. The implicit function method is applied to find some degenerate solutions under constraints on the model parameters. Among the solutions obtained there are doubly periodic solutions on the complex plane, as well as optical solitons.

First integral blow-up solutions to complex-valued KdV equations

We apply Lie theory to determine the aone-parameter point transformations which leave complex-valued Korteweg-de Vries equations invariant. The conserved vectors of the systems are constructed. We provide travelling wave reductions that lead to third-order ordinary differential equations. These equations are highly nonlinear to solve directly and, therefore, we establish their first integrals. The latter is of second-order and facilitates the analysis of the complex-valued system, as well as some interesting blow-up solutions.

Perturbed traveling wave solutions of the CDGKS equation and its dynamics characteristics

Based on the traveling wave reduction method with a perturbed initial solution and the F-expansion method, a class of explicit exact solutions of the (2+1)-dimensional CDGKS equation are obtained through the symbolic computation. Moreover, both the interaction behavior between parameters and the perturbation degree of periodic wave and Gauss wave to rational pulse wave, and the correlation of parameters to the superposition degree of the interaction energy between solitary wave and rational pulse wave are discussed. Finally, numerical simulations are shown to demonstrate the mechanism of the above solutions.

Symmetries and solutions for the inviscid oceanic Rossby wave equation

The -and -dimensional inviscid Rossby wave equations are analysed using Lie symmetry techniques. The travelling-wave reductions for the equation lead to a third-order ordinary differential equation from which the propagation properties are derived. It is observed that the wave has easterly phase velocity and westerly group velocity. Also, the wave propagates slightly faster in the f-plane than the β-plane. The dispersion relation derived from the third-order equation shows that the Rossby equation transports energy both eastward and westward and its speed is reduced successively and the propagation remains eastward. As for the two-dimensional Rossby wave equation, certain solutions which behave like solitary waves after a certain time are plotted. Interestingly, for certain symmetries the reductions lead to the Riccati's, Abel's, Euler's type and to some linearized equations. Moreover, certain reductions lead to highly nonlinear equations which are analysed by the singularity analysis method.

Analytic solutions of the long-wave-short-wave resonance system in fluid mechanics

Under investigation in this paper is the long-wave-short-wave resonance system, which can describe a variety of nonlinear wave phenomena such as the two-dimensional packets of capillary-gravity waves in hydrodynamics and the optical-terahertz waves. The intended aim is carried out via considering a traveling wave reduction, adopting a modified version of the Jacobi elliptic expansion method and employing the Weierstrass elliptic function method to derive such analytic solutions as the bright and dark soliton solutions, periodic solutions, trigonometric-function and elliptic-function solutions in fluid mechanics.

Optical Solitons of the Generalized Nonlinear Schrödinger Equation with Kerr Nonlinearity and Dispersion of Unrestricted Order

The family of the generalized Schrödinger equations with Kerr nonlinearity of unrestricted order is considered. The solutions of equations are looked for using traveling wave reductions. The Painlevé test is applied for finding arbitrary constants in the expansion of the general solution into the Laurent series. It is shown that the equation does not pass the Painlevé test but has two arbitrary constants in local expansion. This fact allows us to look for solitary wave solutions for equations of unrestricted order. The main result of this paper is the theorem of existence of optical solitons for equations of unrestricted order that is proved by direct calculation. The optical solitons for partial differential equations of the twelfth order are given in detail.

Optical Bullets and Their Modulational Instability Analysis

The current work is devoted to investigating the multidimensional solitons known as optical bullets in optical fiber media. The governing model is a (3+1)-dimensional nonlinear Schrödinger system (3D-NLSS). The study is based on deriving the traveling wave reduction from the 3D-NLSS that constructs an elliptic-like equation. The exact solutions of the latter equation are extracted with the aid of two analytic approaches, the projective Riccati equations and the Bernoulli differential equation. Upon applying both methods, a plethora of assorted solutions for the 3D-NLSS are created, which describe mixed optical solitons having the profiles of bright, dark, and singular solitons. Additionally, the employed techniques provide several kinds of periodic wave solutions. The physical structures of some of the derived solutions are depicted to interpret the nature of the medium characterized by the 3D-NLSS. In addition, the modulation instability of the discussed model is examined by making use of the linear stability analysis.

Lie symmetry analysis, soliton solutions and qualitative analysis concerning to the generalized q-deformed Sinh-Gordon equation

In this manuscript, the Lie point symmetries, conservation laws, and traveling wave reductions have all been derived. Also, new forms of soliton solutions of generalized q-deformed equation via means of unified method has been extracted. Implementing this approach the results of governing model are obtained as rational and polynomial functions. The unified technique is used to analyze solitary solutions, soliton solutions, also periodic rational solutions. Additionally, to illustrate the various solitons’ propagation characteristics, several illustrations are used to display their statistical data. The bifurcation is introduced on the governing equation. The planer system has been extracted. Taking all possible cases for the parameters, the behavior has been shown by portraits in phase plane. The proposed equation has expanded modeling possibilities for complex processes with broken symmetry.

Exact solutions of equation for description of embedded solitons

Objective:The generalized Schrödinger equation for description of embedded solitons is considered. The objective of this paper is to find the analytical solutions of the equation. Method:The Cauchy problem for studied partial differential equation cannot be solved by the inverse scattering transform and some exact solutions of equation are looked for using the traveling wave reduction. The system of equations corresponding to the partial differential equation is analyzed. Method of simplest equations is used to find exact solutions of overdetermined system of equations. Results:Exact solutions of the studied equation in the form of periodic and solitary waves are found for one of the equations for description of embedded solitons.

Meromorphic solutions of autonomous ordinary differential equations without the finiteness property

We consider the problem of finding meromorphic solutions of autonomous algebraic ordinary differential equations possessing an infinite number of local solutions given by Laurent series in neighborhoods of poles. We present an algebraic-geometric method for finding all meromorphic solutions that are elliptic or rational in the exponential function. As an example, we classify such families of meromorphic solutions for fourth-order ordinary differential equations arising via integrating once traveling-wave reductions of the generalized Rosenau–Korteweg–de Vries equation and the fifth-order Korteweg–de Vries–Burgers equation. Solutions important from a physical point of view are presented.

Exact solutions of the complex Ginzburg–Landau equation with law of four powers of nonlinearity

The complex Ginzburg–Landau equation having polynomial law of nonlinearity with four powers is considered. The Cauchy problem for this equation cannot be solved by the inverse scattering transform. However the partial differential equation admits the translation groups in two independent variables and we look for the general solution using the traveling wave reduction. The first integral of nonlinear ordinary differential equation corresponding to the complex Ginzburg–Landau equation with the four powers of nonlinearity is found. This first integral is reduced to the nonlinear ordinary differential equation of the first order with the general solution expressed via the elliptic function. We demonstrate that the direct method allows us to obtain exact solutions without constraints on the parameters of the mathematical model. Partial cases of bright and dark optical solitons of the equation are given.

Lie symmetries and exact solutions for a fourth‐order nonlinear diffusion equation

In this paper, we consider a fourth‐order nonlinear diffusion partial differential equation, depending on two arbitrary functions. First, we perform an analysis of the symmetry reductions for this parabolic partial differential equation by applying the Lie symmetry method. The invariance property of a partial differential equation under a Lie group of transformations yields the infinitesimal generators. By using this invariance condition, we present a complete classification of the Lie point symmetries for the different forms of the functions that the partial differential equation involves. Afterwards, the optimal systems of one‐dimensional subalgebras for each maximal Lie algebra are determined, by computing previously the commutation relations, with the Lie bracket operator, and the adjoint representation. Next, the reductions to ordinary differential equations are derived from the optimal systems of one‐dimensional subalgebras. Furthermore, we study travelling wave reductions depending on the form of the two arbitrary functions of the original equation. Some travelling wave solutions are obtained, such as solitons, kinks and periodic waves.

Some exact wave solutions of nonlinear partial differential equations by means of comparison with certain standard ordinary differential equations

We consider applications of a comparison method for obtaining some exact solutions of nonlinear partial differential equations (PDEs) through their traveling wave reductions. The method, proposed by N. A. Kudryashov, 39(18), 5733–5742, Applied Mathematical Modelling, is extended to encompass not only solutions expressible in terms of the logistic function but also to the ‐class of functions in a unified manner. The standard set of second‐order ordinary differential equations (ODEs) admitting the logistic and functions as solutions is derived and extensions to the third‐order case are enunciated. A number of examples are included to illustrate the method.

Highly dispersive optical solitons of the sixth-order differential equation with arbitrary refractive index

Partial differential equation of the sixth order for description of the propagation impulse in nonlinear optics is considered at arbitrary refractive index. Exact solution of differential equation is looked for using the traveling wave reductions. The first integral of nonlinear differential equation is presented. Highly dispersive optical solitons of the equation with arbitrary refractive index is found.

Optical solitons of the model with generalized anti-cubic nonlinearity

The mathematical model for description of pulse propagation in optical fiber for the generalized anti-cubic nonlinearity with arbitrary refractive index is considered. The Cauchy problem for this partial differential equation cannot be solved by the inverse scattering transform and the analytical solutions of this mathematical model are found taking into account the traveling wave reduction. First integral of nonlinear ordinary differential equation is presented. Analytical solutions in the form of periodic and solitary waves are found using some transformations. Partial cases of the mathematical model for the generalized anti-cubic nonlinearity are studied.

Dynamical properties of the periodically perturbed Triki–Biswas equation

In this paper, a perturbed traveling wave reduction of the Tricky–Biswas equation, which is used to describe the propagation of pulses in nonlinear optics, is considered. A stability analysis of the investigated ODE system without perturbation is carried out. The Melnikov function along the homoclinic orbit is constructed. It is found that in the studied system the necessary condition for occurrence of homoclinic chaos is always satisfied. A perturbation is added to the system to control the chaos obtained. Constraints on the parameters of the new system, at which homoclinic chaos is realized in it, are found. The attraction basins are plotted. It is found that their structure is fractal when the damping parameter values are less than the critical ones obtained by the Melnikov approach. The results of the numerical analysis go in agreement with those acquired theoretically.

The travelling wave solutions of nonlinear evolution equations with both a dissipative term and a positive integer power term

<abstract> <p>The formula of solution to a nonlinear ODE with an undetermined coefficient and a positive integer power term of dependent variable have been obtained by the transformation of dependent variable and $(\frac{{G'}}{G})$-expansion method. The travelling wave reduction ODEs (perhaps, after integration and identical deformation) of a class of nonlinear evolution equations with a dissipative term and a positive integer power term of dependent variable that includes GKdV-Burgers equation, GKP-Burgers equation, GZK-Burgers equation, GBoussinesq equation and GKlein-Gordon equation, are all attributed to the same type of ODEs as the nonlinear ODE considered. The kink type of travelling wave solutions for these nonlinear evolution equations are obtained in terms of the formula of solution to the nonlinear ODE considered.</p> </abstract>