SOLITARY WAVES OF THE HIERARCHY EQUATIONS BURGERS

Cover Page

Cite item

Full Text

Open Access Open Access
Restricted Access Access granted
Restricted Access Subscription Access

Abstract

Burgers hierarchy equations are considered. It is shown that the well-known Cole-Hopf transformation for linearization of the classical Burgers equation generalizes to the case of equations of arbitrary order of the Burgers hierarchy. This fact allows us to find solitary and periodic waves described by the Burgers hierarchy equations, resembling the N-wave for the classical Burgers equation. A detailed consideration of the construction of solitary waves is presented for the third-order Sharma-Tasso-Olver equation and for the fourth-order hierarchy equation. It is found that for the third-order dissipative equation, N-wave type solitary waves have oscillations at the solution front. In the case of second and fourth order dissipative equations such oscillations are absent.

About the authors

N. A. Kudryashov

National Research Nuclear University MEPhI

Email: NAKudryashov@mephi.ru
Moscow, Russia

References

  1. Ablowitz M.J., Clarkson P.A. Solitons Nonlinear Evolution Equations and Inverse Scattering. Cambridge: Cambridge University Press, 1991.
  2. Drazin P.G., Johnson R.S. Solitons: An Introduction. Cambridge: Cambridge University Press, 1989.
  3. Агравал Г. Нелинейная волоконная оптика. М.: Мир, 1996.
  4. Кившарь Ю.С., Агравал Г. Оптические солитоны. От волоконных световодов к фотонным кристаллам. М.: Физматлит, 2005.
  5. Кудряшов Н.А. Методы нелинейной математической физики. Долгопрудный: Интеллект, 2019.
  6. Polyanin A.D., Zaitsev V.F. Handbook of Nonlinear Partial Differential Equations, 2nd ed. Boca Raton: CRC Press, 2012.
  7. Weiss J., Tabor M., Carnevale G. The Painleve property for partial differential equations // J. Math. Phys. 1982. V. 24. №3. P. 522–526.
  8. Gardner C.S., Greene J.M., Kruskal M.D., Miura R.M. Method for solving the Korteweg-deVries equation // Physical Review Letters. 1967. V. 19.№19. P. 1095–1097.
  9. Lax P.D. Integrals of nonlinear equations of evolution and solitary waves // Communications on Pure and Applied Mathematics. 1968. V. 21.№5. P. 467–490.
  10. Kaup D.J., Newell A.C. An exact solution for a derivative Schr¨odinger equation // J. of Math. Physics. 1979. 19. 798.
  11. Куликовский А.Г., Чугайнова А.П., Шаргатов В.А. Единственность автомодельных решений задачи о распаде произвольного разрыва уравнения Хопфа со сложной нелинейностью // Ж. вычисл. матем. и матем. физ. 2016. V. 56.№7. P. 1363–1370.
  12. Куликовский А.Г., Чугайнова А.П. Исследование разрывов в решениях уравнений упругопластической среды Прандтля-Рейсса // Ж. вычисл. матем. и матем. физ. 2016. V. 56.№4. P. 650–66.
  13. Куликовский А.Г., Ильичев А.Т., Чугайнова А.П., Шаргатов В.А. О спонтанно излучающих ударных волнах. Докл. АН. 2019. 487.№1. С. 28–31.
  14. Chugainova A.P., Kulikovskii A.G. Shock waves in an incompressible anisotropic elastoplastic medium with hardening and their structures // Appl. Math. and Comput. 2021.№401. 126077.
  15. Bateman, H. Some Recent Researches on the Motion of Fluids. MonthlyWeather Review. 1915.№43. P. 163–170. http://dx.doi.org/10.1175/1520-0493(1915)43<163:SRROTM>2.0.CO;2.
  16. Burgers J.M. A mathematical model illustrating the theory turbulence // Adv. Appl. Mech. 1948.№1. P. 171–199.
  17. Whitham G.B. Linear and nonlinear waves, A Wiley-interscience Publication? New York: J. Wiley and Sons, 1974.
  18. Billingham J., King A.C.. Wave Motion. Cambridge University press. 2000.№468.
  19. Cole J.D. On a quasi-linear parabolic equation occuring in aerodynamics // Quart. Appl. Math. 1951. V. 9. № 3. P. 225–236.
  20. Hopf E. The partial differential equation ut + uux = uxx // Commun. Pure Appl. Math. 1950. V. 3.№3. P. 201–230.
  21. Sharma AS, Tasso H. Connection Between Wave Envelope and Explicit Solution of a Nonlinear Dispersive Wave Equation. Muenchen, Germany: Max-Planck-Institut f ¨ur Plasmaphysik. 1976.
  22. Olver J.P. Evolution equations posessing infinitely many symmetries // J. Math. Phys. 1977. V. 18.№6. P. 1212–1215.
  23. Johnpillai, Andrew Gratien and Khalique, Chaudry Masood. On the solutions and conservation laws for the SharmaTasso-Olver equation. ScienceAsia. 2014. V. 40.№6. P. 451–455. https://doi.org/10.2306/scienceasia1513-1874.2014.40.451.
  24. Khan, Mohammad Jibran and Nawaz, Rashid and Farid, Samreen and Iqbal, Javed. New iterative method for the solution of fractional damped burger and fractional sharma-tasso-olver equations. Complexity. 2018. https://doi.org/10.1155/2018/3249720.
  25. Ren, Bo and Ma, Wen-Xiu. Rational solutions of a (2+1)-dimensional Sharma-Tasso-Olver equation // Chinese Journal of Physics. 2019.№60. P. 153–157. https://doi.org/10.1016/j.cjph.2019.05.004.
  26. Roy, Ripan and Akbar, M. Ali and Wazwaz, Abdul Majid. Exact wave solutions for the nonlinear time fractional Sharma–Tasso–Olver equation and the fractional Klein–Gordon equation in mathematical physics // Optical and Quantum Electronics. 2018. V. 50.№1. https://doi.org/10.1007/s11082-017-1296-9.
  27. Zhou, Tian-Yu and Tian, Bo and Chen, Yu-Qi. Elastic Two-Kink, Breather, Multiple Periodic, Hybrid and Half/Local-Periodic Kink Solutions of a Sharma-Tasso-Olver-Burgers Equation for the Nonlinear Dispersive Waves // Qualitative Theory of Dynamical Systems 2023. V. 22.№1. https://doi.org/10.1007/s12346-022-00713-8.
  28. Zayed, Elsayed M.E. and Amer, Yasser A. and Shohib, Reham M.A.. The fractional expansion method and its applications for solving four nonlinear space-time fractional PDES in mathematical physics // Italian Journal of Pure and Applied Mathematics. 2015.№34. P. 463–482.
  29. Chen, Cheng and Jiang, Yao-Lin. Simplest equation method for some time-fractional partial differential equations with conformable derivative // Computers and Mathematics with Applications. 2018. V. 75.№8. P. 2978–2988. https://doi.org/10.1016/j.camwa.2018.01.025.
  30. Miao, Zhengwu and Hu, Xiaorui and Chen, Yong. Interaction phenomenon to (1+1)-dimensional Sharma–Tasso–Olver–Burgers equation // Applied Mathematics Letters. 2021.№112. https://doi.org/10.1016/j.aml.2020.106722.
  31. Kumar, Sachin and Khan, Ilyas and Rani, Setu and Ghanbari, Behzad. Lie Symmetry Analysis and Dynamics of Exact Solutions of the (2+1)-Dimensional Nonlinear Sharma-Tasso-Olver Equation // Mathematical Problems in Engineering. 2021. https://doi.org/10.1155/2021/9961764.
  32. Wang, Kang-Jia and Shi, Feng and Liu, Jing-Hua. A Fractal Modification Of The Sharma-TassoOlver Equation and Its Fractal Generalized Variational Principle. Fractals. 2022. V. 30. № 6. https://doi.org/10.1142/S0218348X22501213.
  33. Yassin, Ola and Alquran, Marwan. Constructing new solutions for some types of two-mode nonlinear equations // Applied Mathematics and Information Sciences. 2018. V. 12. № 2. P. 361–367. https://doi.org/10.18576/amis/120210.
  34. Ghose-Choudhury, A. and Garai, Sudip. Some exact wave solutions of nonlinear partial differential equations by means of comparison with certain standard ordinary differential equations // Mathematical Methods in the Applied Sciences. 2022. V. 45.№16. P. 9297–9307. https://doi.org/10.1002/mma.8305.
  35. Sirisubtawee, Sekson and Koonprasert, Sanoe and Sungnul, Surattana. New exact solutions of the conformable spacetime Sharma-Tasso-Olver equation using two reliable methods // Symmetry. 2020. V. 12.№4. https://doi.org/10.3390/SYM12040644.
  36. Kai, Yue and Yin, Zhixiang. Linear structure and soliton molecules of Sharma-Tasso-Olver-Burgers equation. Physics Letters, Section A: General, Atomic and Solid State Physics. 2022.№452. https://doi.org/10.1016/j.physleta.2022.128430.
  37. Kudryashov, N.A.. Self-similar solutions of the Burgers hierarchy // Applied Mathematics and Computation. 2009. V. 215.№5. P. 1990–1993. https://doi.org/10.1016/j.amc.2009.07.048.
  38. Kudryashov, N.A.. Generalized Hermite polynomials for the Burgers hierarchy and point vortices // Chaos, Solitons and Fractals. 2021.№151. https://doi.org/10.1016/j.chaos.2021.111256.
  39. Kudryashov, N.A. and Sinelshchikov, D.I.. The Cauchy problem for the equation of the Burgers hierarchy // Nonlinear Dynamics. 2014. V. 76.№1. P. 561–569. https://doi.org/10.1007/s11071-013-1149-4.

Supplementary files

Supplementary Files
Action
1. JATS XML
Download

Copyright (c) 2025 Russian Academy of Sciences