Search

Analytical solutions of the generalized nonlinear Schrödinger with four powers of nonlinearity for description of propagating pulses in optical fiber are presented. Optical solitons corresponding to the mathematical model are given. Conservation laws of the generalized model for propagation pulses with four powers of nonlinearity are written. To the best of our knowledge, the conservation laws obtained have not yet been presented in literature. The equation investigated generalizes several well-known models, which allows us to evaluate the influence of various processes on pulse propagation. Conservative quantities for the bright optical soliton, corresponding to its power, momentum and energy, are calculated. The analytical expressions for conservative quantities obtained can be applied to check whether numerical schemes for the explored equation are conservative. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

This article explores the generalized Gerdjikov–Ivanov equation describing the propagation of pulses in optical fiber. The equation studied has a variety of applications, for instance, in photonic crystal fibers. In contrast to the classical Gerdjikov–Ivanov equation, the solution of the Cauchy problem for the studied equation cannot be found by the inverse scattering problem method. In this regard, analytical solutions for the generalized Gerdjikov–Ivanov equation are found using traveling-wave variables. Phase portraits of an ordinary differential equation corresponding to the partial differential equation under consideration are constructed. Three conservation laws for the generalized equation corresponding to power conservation, moment and energy are found by the method of direct transformations. Conservative densities corresponding to optical solitons of the generalized Gerdjikov–Ivanov equation are provided. The conservative quantities obtained have not been presented before in the literature, to the best of our knowledge. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

The fourth-order nonlinear Schrödinger equation with four power nonlinearities is considered. The equation studied is the generalization of some well-known models and allows us to evaluate the influence various processes of pulse propagation. The main property of this equation is not integrability because the Cauchy problem for it cannot be solved by the inverse scattering transform. However this equation has some analytical solutions. Optical and embedded solitons corresponding to the considered mathematical model are given. Conservation laws of the partial differential equation for propagation pulses are found. Conservative quantities for the bright and embedded optical solitons are calculated. [ABSTRACT FROM AUTHOR]

Traveling wave reduction of the Kundu–Mukherjee–Naskar model for describing of the optical solitons is considered. Two first integrals for the system of equations are found. These first integrals are used to obtain the nonlinear second-order differential equation. The general solution of the second-order ordinary differential equation is found via the Weierstrass and Jacobi elliptic functions. [ABSTRACT FROM AUTHOR]

The traveling wave reduction of the Schrödinger equation with anti-cubic nonlinearity for describing of the optical solitons is studied. We find two first integrals of the system of equations corresponding to the real and imaginary part of the optical pulse. Using the first integrals we obtain the general and special solutions of the Schrödinger equation with anti-cubic nonlinearity. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

Conservation laws of the generalized nonlinear Schrödinger equation with arbitrary refractive index of the fourth order are found. Conservative quantities corresponding to the bright optical soliton are calculated. Direct calculations of three conservation laws for the considered equation are used. This approach allows us to obtain the conservative densities and fluxes of the equation. It is shown that there exist three conservation laws corresponding to the studied equation. The bright optical solitons of the equation are obtained. Using these solutions, conservative quantities of the soliton for the equation are calculated by means of integration. Some properties of conservation laws are discussed. [ABSTRACT FROM AUTHOR]

The generalized Schrödinger–Hirota equation of the fourth order is considered using the traveling wave solutions. The compatibility of the overdetermined system of equations is analyzed. The Painleé test is applied to determine the integrability property of the mathematical model. The simplest equation method is used for finding periodic and solitary wave solutions. Optical solitons of the generalized Schrödinger–Hirota equation of the fourth order are found. [ABSTRACT FROM AUTHOR]

The Generalized Schrödinger–Hirota equation with an arbitrary power of nonlinearity is studied. The compatibility of the overdetermined system of equation is analyzed. The task is reduced to studying one of the equations which is solved taking into account the direct calculations by means of variable transformations. Dispersive optical solitons and periodic solutions of the generalized Schrödinger–Hirota equation with arbitrary power of nonlinearity are found. [ABSTRACT FROM AUTHOR]

This paper conducts Painlevé analysis to a concatenated model that stems from the well-known nonlinear Schrödinger's equation, Lakshmanan–Porsezian–Daniel model and Sasa–Satsuma equation. A bright 1-soliton solution is subsequently recovered through an intermediary Jacobi's elliptic function. [ABSTRACT FROM AUTHOR]

The mathematical model having unrestricted dispersion and nonlinearity in the polynomial form of described by the nonlinear Schrödinger equation is considered. Rational solitary wave solutions and optical solitons of the nonlinear partial differential equation are looked for. The Painlevé test is applied to analyze the integrability of the nonlinear partial differential equation. Two variants of the simplest equation method are used for finding rational solitary waves and optical solitons of the mathematical model. It is shown that the mathematical model having unrestricted dispersion and nonlinearity in the form of polynomial is nonintegrable by the inverse scattering transform. It is proved the general solution of the nonlinear ordinary differential equation has the expansion of in the Laurent series with two arbitrary constants. Rational solitary wave solutions and optical solutions of the mathematical model are found. [ABSTRACT FROM AUTHOR]

The generalized Schrödinger equation with arbitrary refractive index and unrestricted order is considered. The objective of this paper is to demonstrate the bright solitons of this mathematical model can be found by the same formula taking into account the order of nonlinear differential equation. The algorithm of constructing optical solitons of the generalized nonlinear Schrödinger equation with unrestricted dispersion using the simplest equation method is presented and applied for finding of optical solitons. The optical solitons of the generalized nonlinear Schrödinger equation of the twelfth order are found at three powers on nonlinearity. The application of the method for finding solution of the generalized Korteweg–de Vries equation with arbitrary power of nonlinearity and unrestricted dispersion is discussed. [ABSTRACT FROM AUTHOR]

Partial differential equation in a dual-power law nonlinear medium with variable coefficients is considered. The objective is to find solitary wave solution that can be considered as a governed optical solitons. Exact solutions of nonlinear differential equation are found using the direct method. Constraints on variables of coefficients are used for reducing equation to the nonlinear ordinary differential equation having analytical solutions. Periodic and solitary wave solutions are obtained of nonlinear partial differential equation with dual-power law of nonlinearity and coefficients depending on variable. Two simple examples of acceleration and deceleration of solitary wave solutions are discussed. [ABSTRACT FROM AUTHOR]

New generalization of the nonlinear Schrödinger equation with triple refractive index and non-local nonlinearity is considered and new form of optical solitons is given. An auxiliary nonlinear differential equation of the first order with a double nonlinearity is introduced in order to find exact solutions of a nonlinear partial differential equation. Exact solutions of this auxiliary equation is presented. It is shown that there are new type optical solitons of the generalized Schrödinger equation that are expressed via the implicit functions. [ABSTRACT FROM AUTHOR]

New generalized Schrödinger equations with polynomial nonlinearities are considered. The Cauchy problem for these equations cannot be solved by the inverse scattering transform and optical solitons of these equations are looked for taking into account the traveling wave solutions. Application of the well-known auxiliary equations as the Riccati equation and equations for elliptic functions for construction of solutions of new generalized Schrödinger equations is impossible right away. Therefore solutions of nonlinear ordinary differential equations are found using the transformations of dependent and independent variables. This extended approach allows us to obtain some new auxiliary nonlinear ordinary equations. New auxiliary differential equations allow to look for optical solitons of the other generalized Schrödinger equations. We demonstrate that by using new auxiliary equations, we can find the optical solitons of the generalized nonlinear Schrödinger equations of the fourth degree with a polynomial of the eighteenth power. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

The generalized nonlinear Schrödinger equation with nonlinear chromatic dispersion and polynomial of powers with arbitrary refractive index is considered. The first integral of the nonlinear ordinary differential equation corresponding to the stationary solution of equation is presented. Analytical solutions for bright and dark stationary optical solitons described by the generalized mathematical model are found using transformations of variables. [ABSTRACT FROM AUTHOR]

Stationary optical solitons described by the generalized nonlinear Schrödinger equation are studied. This generalized nonlinear Schrödinger equation takes into account nonlinear chromatic dispersion and expressions with two arbitrary reflection indices. It can be used for description of propagation pulses in nonlinear optics. Analytical expressions for stationary solitons are found using transformations for dependent and independent variables. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

• Generalized nonlinear Schrödinger equation for description of optical solitons in a saturable medium is studied. • The transformation variables are used for finding solitary wave solutions. • Bright and dark solitons for the description are found in the form of implicit functions. The generalized nonlinear Schrödinger equation for description of optical solitons in a saturable medium is studied. The transformation variables are used for finding solitary wave solutions. Optical solitons for the description of pulse propagation are obtained in the form of implicit functions. Analytical solutions of the generalized nonlinear Schrödinger equation obtained for bright and dark solitons in a saturable medium are demonstrated. [ABSTRACT FROM AUTHOR]

The generalized nonlinear Schrödinger equation with two arbitrary reflective indices is considered. Application of transformations for dependent and independent variables is used for finding solitary wave solutions. Parametric optical solitons for description of propagation pulse in nonlinear optics are found for some condition of mathematical models. [ABSTRACT FROM AUTHOR]

The perturbed Chen-Lee-Liu equation with arbitrary refractive index is studied. The traveling wave reduction is used to look for the solution of this partial differential equation. The compatibility conditions for the system of ordinary differential equations are determined. Exact solutions of mathematical model are found. They are expressed via the Jacobi and the Weierstrass elliptic functions. Optical solitons are found in the case of arbitrary refractive index. Application of the Melnikov criterion for evaluation of dynamical chaos at propagation of solitary waves in an optical fiber is discussed. [ABSTRACT FROM AUTHOR]

The generalized Lakshmanan–Porsezian–Daniel model with arbitrary refractive index is considered. The main problem that is solved for this nonlinear partial differential equation is the existence of exact solutions in the form of periodic and solitary waves. It is shown that for an arbitrary refractive index of the equation, there are solutions in the form of solitary waves corresponding to optical solitons. For the classical case of the Lakshmanan–Porsezian–Daniel equation at n = 1 and for new case at n = 1 2 the equation has a solution in the form of periodic waves. These exact solutions are expressed via the Jacobi and the Weierstrass elliptic functions. [ABSTRACT FROM AUTHOR]

Traveling wave reduction of the resonant nonlinear Schrödinger equation with arbitrary refractive index is considered. The system of equations for real and imaginary parts is presented. The first integrals for the system of equations are found. This system of equations is transformed to the only governing nonlinear ordinary differential equations of the first order. Exact solutions in the form of periodic and solitary waves of this nonlinear ordinary differential equation are given for various values of the refractive index. [ABSTRACT FROM AUTHOR]

A new generalization of the nonlinear Schrödinger equation is presented. A characteristic feature of this equation is that it takes into account both an arbitrary reflective index in optical fiber and the phenomenon of non-local nonlinearity. It is shown that the equation has exact solutions in the form of solitary waves, which can be considered as optical solitons. Special cases of the equation and its solutions are also considered. [ABSTRACT FROM AUTHOR]

The mathematical model with arbitrary refractive index is considered. The equation studied is the generalization of the Schrödinger equation with cubic and quintic nonlinearity. Optical solitons of equation are found taking into account the traveling wave reduction. [ABSTRACT FROM AUTHOR]

A mathematical model for describing the propagation of pulses in an optical fiber with an arbitrary refractive index is considered. In order to find a solution to the differential equation, we used the hypothesis of a possible solution, which, combined with the method of simplest equation, allowed to find optical solitons this new nonlinear partial differential equation. [ABSTRACT FROM AUTHOR]

Hierarchy of the perturbed nonlinear Schrödinger equations is considered. Nonlinear differential equations of this hierarchy contain higher orders and can be used for description of highly dispersive optical solutions. A new approach for finding solitary wave solutions of high-order nonlinear differential equations is presented. This approach allows us to significantly simplify symbolic calculations. The main idea of the method is that we use expressions of the dependent variable and its derivatives in the differential equation the polynomial form of the solitary wave. We find optical solitons with high dispersion order for nonlinear perturbed Schrodinger equations of the fourth, sixth, eighth, tenth and twelfth orders. [ABSTRACT FROM AUTHOR]

The generalized nonlinear eighth-order Schrödinger equation with third, fifth, seventh and ninth power of nonlinearity is studied. This equation can be used for description of the pulse propagation in optical fiber. It is demonstrated that this equation has highly dispersive optical solitons which are presented in the analytical form. [ABSTRACT FROM AUTHOR]

A method for finding solitary wave solutions to nonlinear differential equations is presented. A generalization for the logistic function to obtain a solitary wave solution is introduced. Properties of this basic function are discussed. An algorithm for finding exact solutions in the form of a solitary wave for nonlinear differential equations is formulated. The method has significant advantages over other approaches of this type. Its advantage is due to the fact that in the calculations we do not use the form of a specific function. Our approach is particularly effective in finding exact solutions to high-order nonlinear differential equations used in describing the propagation of pulses in an optical fiber. The application of the exact solutions search method for finding highly dispersed solitons of nonlinear differential equations is shown. [ABSTRACT FROM AUTHOR]