Персона: Кудряшов, Николай Алексеевич
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Traveling wave solutions of the generalized Gerdjikov–Ivanov equation
© 2020 Elsevier GmbHThe traveling wave solutions of the generalized Gerdjikov–Ivanov equation are studied. We find the first integrals for the system of equations corresponding to imaginary and real parts to this equation. The general solution with three arbitrary constants is found which are expressed via the elliptic functions. Periodic and solitary wave solutions of the generalized Gerdjikov–Ivanov equation are illustrated for some values of parameters.
Точные решения обобщенного нелинейного уравнения Вахненко – Паркеса
Рассматривается одно из уравнений семейства обобщенных уравнений Вахненко – Паркеса, описывающих распространение коротковолновых возмущений в релаксирующих средах в случае, когда амплитуда колебаний зависит от скорости распространения волны. Для данного уравнения получено общее решение, записанное через квадратуру, путем сведения его к обыкновенному дифференциальному уравнению второго порядка с использованием переменных бегущей волны. Исследовано влияние параметров уравнения на полученное решение. Найдены его точные решения. Периодические точные решения выражены через эллиптические функции Якоби. Кроме того, представлено явное решение, выражаемое через степенную функцию пространственной и временной переменных. Полученные точные решения могут быть использованы в качестве тестовых функций при анализе результатов численного моделирования процессов в релаксирующих средах, описываемых уравнениями типа Вахненко – Паркеса.
On general solutions of two nonlinear ordinary differential equations
© 2019 Author(s).A generalization of the simplest equation method for finding exact solutions is presented. The basic idea of the method is to use simplest equations with function that is found during the calculations. Using our algorithm we consider two types of nonlinear ordinary differential equations of the second and the third order. The method is applied for finding exact solutions and first integrals. The exact solutions of some nonlinear differential equations are found. It is shown that the most methods that were used to construct exact solutions for differential equations are included in suggested approach.
Painleve analysis and traveling wave solutions of the fourth-order differential equation for pulse with non-local nonlinearity
© 2020 Elsevier GmbHNonlinear fourth-order partial differential equation with non-local nonlinearity for describing pulses in optical fiber is considered. The traveling wave reductions to the equation are used to obtain the real and imaginary parts of nonlinear differential equation. Using the Painlevé analysis to the system of equations it is shown that this system does not have the general solution with four arbitrary constants. However the equation can have exact solution with the smaller number of arbitrary constants. Conditions for some parameters of the mathematical model are found for solution of the system of equations. Exact solutions for the system of equations are found by the means of the simplest equation method. Exact solutions are given using the Jacobi elliptic functions.
On Specific Features of an Approach Based on Feedforward Neural Networks to Solve Problems Based on Differential Equations
Solitary and periodic waves of the hierarchy for propagation pulse in optical fiber
© 2019 Elsevier GmbHWe consider the hierarchy of partial differential equations with arbitrary power-low nonlinearity which can be used for description of the propagation pulse in optical fiber. The Cauchy problem for these partial differential equations cannot be solved by the inverse scattering transform and we look for exact solutions of differential equations using the traveling wave reduction. It is proven that all equations of the hierarchy have exact solutions in the form of periodic and solitary waves that are determined by means of the elliptic functions. A more detailed study of the hierarchy is presented for the equations of the second, fourth and sixth order. The parameter values for existence of exact solutions for these equations are given. Exact solutions of differential equations are expressed in terms of the Weierstrass elliptic function. A formula for describing solitary waves is also given. Exact solutions in the form of periodic and solitary waves for differential equations are illustrated.
Embedded Solitons of the Generalized Nonlinear Schrodinger Equation with High Dispersion
The family of generalized Schrödinger equations is considered with the Kerr nonlinearity. The partial differential equations are not integrable by the inverse scattering transform and new solutions of this family are sought taking into account the traveling wave reduction. The compatibility of the overdetermined system of equations is analyzed and constraints for parameters of equations are obtained.A modification of the simplest equation method for finding embedded solitons is presented.A block diagram for finding a solution to the nonlinear ordinary differential equation isgiven. The theorem on the existence of bright solitons for differential equations of any orderwith Kerr nonlinearity of the family considered is proved. Exact solutions of embedded solitonsdescribed by fourth-, sixth-, eighth and tenth-order equations are found using the modified algorithm of the simplest equation method. New solutions for embedded solitons of generalized nonlinear Schrödinger equations with several extremes are obtained. © 2022, Pleiades Publishing, Ltd.
Optical solitons of the Chen–Lee–Liu equation with arbitrary refractive index
© 2021 Elsevier GmbHThe 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.