vestnikmephiВестник НИЯУ МИФИVestnik natsional'nogo issledovatel'skogo yadernogo universiteta "MIFI"2304-487XNational Research Nuclear University "MEPhI"10.1134/S2304487X21040088vestnikmephi-166Research ArticleДИФФЕРЕНЦИАЛЬНЫЕ УРАВНЕНИЯ И ДИНАМИЧЕСКИЕ СИСТЕМЫDIFFERENTIAL EQUATIONS AND DYNAMIC SYSTEMSСолитонные решения обобщенного уравнения Чена–Ли–ЛюSoliton Solutions of the Chen–Li–Liu Equation with an Arbitrary Refractive IndexКудряшовН. А.KudryashovN. A.

115409

Москва

115409

Moscow

nakudr@gmail.comЕрмолаеваН. В.ErmolaevaN. V.

115409

347360

Москва

Ростовская обл.

Волгодонск

115409

347360

Moscow

Rostovskaya oblast

Volgodonsk

NVErmolayeva@mephi.ruНациональный исследовательский ядерный университет “МИФИ”РоссияNational Research Nuclear University MEPhI (Moscow Engineering Physics Institute)Russian FederationНациональный исследовательский ядерный университет “МИФИ”; Волгодонский инженерно-технический институт НИЯУ МИФИРоссияNational Research Nuclear University MEPhI (Moscow Engineering Physics Institute); Volgodonsk Engineering Technical Institute, National Research Nuclear University MEPhI (Moscow Engineering Physics Institute)Russian Federation202122022023104302307Copyright © Кудряшов Н.А., Ермолаева Н.В., 20232023Кудряшов Н.А., Ермолаева Н.В.Kudryashov N.A., Ermolaeva N.V.Данная работа распространяется под лицензией Creative Commons Attribution 4.0.This work is licensed under a Creative Commons Attribution 4.0 License.https://vestnikmephi.elpub.ru/jour/article/view/166

   Рассматривается возмущенное уравнение Чена–Ли–Лю с произвольным показателем преломления, описывающее распространение импульсов в нелинейной оптике. Для поиска решения данного нелинейного дифференциального уравнения в частных производных используются переменные бегущей волны. Разделяя мнимую и действительные части полученного уравнения и приравнивая их к нулю, получена система обыкновенных дифференциальных уравнений (ОДУ). Определены условия совместимости системы ОДУ. Найдены стационарные точки системы уравнений. Получены точные решения математической модели при n = 1 и n = 2, выраженные через эллиптические функции Якоби и Вейерштрасса. Показано, что найденные решения в случае произвольного показателя преломления имеют форму периодических и уединенных волн (оптических солитонов).

   The perturbed Chen–Li–Liu equation with an arbitrary refractive index describing the propagation of pulses in an optical fiber is considered. The traveling wave reduction is used to find a solution of this nonlinear partial differential equation. Separating the imaginary and real parts of the resulting equation and equating them to zero, a system of ordinary differential equations is constructed. The compatibility conditions of the system of ordinary differential equations are determined. Stationary points of the system of equations are found. Exact solutions of the mathematical model are obtained for n = 1 and 2 expressed in terms of the Jacobi and Weierstrass elliptic functions. It is shown that the solutions found in the case of an arbitrary refractive index have the form of periodic and solitary waves (optical solitons).

возмущенное уравнение Чена–Ли–Люпроизвольный показатель преломлениябегущая волнаточное решениеоптический солитонperturbed Chen–Li–Liu equationarbitrary refractive indextraveling waveexact solutionoptical solitonРабота поддержана Министерством науки и высшего образования Российской Федерации (государственный проект) № 0723-2020-0036, а также финансировалась Российским фондом фундамен- тальных исследований по исследовательскому проекту № 18-29-10025The work was supported by the Ministry of Science and Higher Education of the Russian Federation (state project) No. 0723-2020-0036, and was also financed by the Russian Foundation research project No. 18-29-10025ReferencesBiswas A. Optical soliton perturbation with Radhakrishnan–Kundu–Lakshmanan equation by traveling wave hypothesis // Optik. 2018. V. 171. P. 217–220.Biswas A. Optical soliton perturbation with Radhakrishnan–Kundu–Lakshmanan equation by traveling wave hypothesis // Optik. 2018. V. 171. P. 217–220.Kudryashov N. A. First integrals and solutions of the traveling wave reduction for the Triki–Biswas equation // Optik. 2019. V. 185. P. 275–281.Kudryashov N. A. First integrals and solutions of the traveling wave reduction for the Triki–Biswas equation // Optik. 2019. V. 185. P. 275–281.Biswas A. Chirp-free bright optical soliton perurbation with Chen–Lee–Liu equation by traveling hypothesis and semi-inverse variational principle // Optik. 2018. V. 172. P. 772–776.Biswas A. Chirp-free bright optical soliton perurbation with Chen–Lee–Liu equation by traveling hypothesis and semi-inverse variational principle // Optik. 2018. V. 172. P. 772–776.Tsuchida T., Wadati M. New integrable systems of derivative nonlinear Schrödinger equations with multiple components // Phys. Lett. A. 1999. V. 257 (1–2). P. 53–64.Tsuchida T., Wadati M. New integrable systems of derivative nonlinear Schrödinger equations with multiple components // Phys. Lett. A. 1999. V. 257 (1–2). P. 53–64.Fan E. A family of completely integrable multi-Hamiltonian systems explicitly related to some celebrated equations // J. Math. Phys. 2001. V. 42 (9). P. 4327–4344.Fan E. A family of completely integrable multi-Hamiltonian systems explicitly related to some celebrated equations // J. Math. Phys. 2001. V. 42 (9). P. 4327–4344.Yang B., Zhang W.-G., Zhang H.-Q., Pei S.-B. Generalized Darboux transformation and rational soliton solutions for Chen–Lee–Liu equation // Appl. Math. Comput. 2014. V. 242. P. 863–876.Yang B., Zhang W.-G., Zhang H.-Q., Pei S.-B. Generalized Darboux transformation and rational soliton solutions for Chen–Lee–Liu equation // Appl. Math. Comput. 2014. V. 242. P. 863–876.Kudryashov N. A. General solution of the traveling wave reduction for the perturbed Chen-Lee-Liu equation // Optik, 2019. V. 186. P. 339–349.Kudryashov N. A. General solution of the traveling wave reduction for the perturbed Chen-Lee-Liu equation // Optik, 2019. V. 186. P. 339–349.Triki H., Babatin M. M., Biswas A. Chirped bright solitons for Chen–Lee–Liu equation in optical fibers and PCF // Optik. 2018. V. 149. P. 300–303.Triki H., Babatin M. M., Biswas A. Chirped bright solitons for Chen–Lee–Liu equation in optical fibers and PCF // Optik. 2018. V. 149. P. 300–303.Triki H., Hamaizia Y., Zhou Q., Biswas A., Ullahd M. Z., Moshokoae S. P., Belic M. Chirped dark and gray solitons for Chen–Lee–Liu equation in optical fibers and PCF // Optik. 2018. V. 155. P. 329–333.Triki H., Hamaizia Y., Zhou Q., Biswas A., Ullahd M. Z., Moshokoae S. P., Belic M. Chirped dark and gray solitons for Chen–Lee–Liu equation in optical fibers and PCF // Optik. 2018. V. 155. P. 329–333.Jawad A. J. M., Biswas A., Zhou Q., Alfiras M., Moshokoa S. P., Belic M. Chirped singular and combo optical solitons for Chen–Lee–Liu equation with three forms of integration architecture // Optik. 2018. V. 178. P. 172–177.Jawad A. J. M., Biswas A., Zhou Q., Alfiras M., Moshokoa S. P., Belic M. Chirped singular and combo optical solitons for Chen–Lee–Liu equation with three forms of integration architecture // Optik. 2018. V. 178. P. 172–177.Kudryashov N. A. A generalized model for description of propagation pulses in optical fiber // Optik. 2019. V. 189. P. 42–52.Kudryashov N. A. A generalized model for description of propagation pulses in optical fiber // Optik. 2019. V. 189. P. 42–52.Kudryashov N. A. Mathematical model of propagation pulse in optical fiber with power nonlinearities // Optik. 2020. V. 212. P. 164750.Kudryashov N. A. Mathematical model of propagation pulse in optical fiber with power nonlinearities // Optik. 2020. V. 212. P. 164750.Kudryashov N. A. Highly dispersive solitary wave solutions of perturbed nonlinear Schrödinger equations // Appl. Math. Comput. 2020. V. 371. P. 124972.Kudryashov N. A. Highly dispersive solitary wave solutions of perturbed nonlinear Schrödinger equations // Appl. Math. Comput. 2020. V. 371. P. 124972.Biswas A., Ekici M., Dakova A., Khan S., Moshokoa S. P., Alshehri H. M., Belic M. R. Highly dispersive optical soliton perturbation with Kudryashov’s sextic-power law nonlinear refractive index by semi-inverse variation // Results Phys. 2021. V. 27. P. 104539.Biswas A., Ekici M., Dakova A., Khan S., Moshokoa S. P., Alshehri H. M., Belic M. R. Highly dispersive optical soliton perturbation with Kudryashov’s sextic-power law nonlinear refractive index by semi-inverse variation // Results Phys. 2021. V. 27. P. 104539.Arrowsmith D. K., Place C. M. Ordinary Differential Equations. A Qualitative Approach with Applications, Chapman and Hall. London, New York, 1982.Arrowsmith D. K., Place C. M. Ordinary Differential Equations. A Qualitative Approach with Applications, Chapman and Hall. London, New York, 1982.Kudryashov N. A. The Lakshmanan–Porsezian–Daniel model with arbitrary refractive index and its solution // Optik. 2021. V. 241. P. 167043.Kudryashov N. A. The Lakshmanan–Porsezian–Daniel model with arbitrary refractive index and its solution // Optik. 2021. V. 241. P. 167043.Kudryashov N. A. Construction of nonlinear differential equations for description of propagation pulses in optical fiber // Optik. 2019. V. 192. P. 162964.Kudryashov N. A. Construction of nonlinear differential equations for description of propagation pulses in optical fiber // Optik. 2019. V. 192. P. 162964.

The authors declare that there are no conflicts of interest present.