RUS  ENG JOURNALS   PEOPLE   ORGANISATIONS   CONFERENCES   SEMINARS   VIDEO LIBRARY   PERSONAL OFFICE
 General information Latest issue Archive Impact factor Subscription Guidelines for authors License agreement Submit a manuscript Search papers Search references RSS Latest issue Current issues Archive issues What is RSS

 TMF: Year: Volume: Issue: Page: Find

 TMF, 2002, Volume 132, Number 1, Pages 90–96 (Mi tmf348)

Exact Solutions for a Family of Variable-Coefficient “Reaction–Duffing” Equations via the Bäcklund Transformation

Yong Chen, Zhenya Yan, Hongqing Zhang

Dalian University of Technology

Abstract: The homogeneous balance method is extended and applied to a class of variable-coefficient “reaction–duffing” equations, and a Bäcklund transformation (BT) is obtained. Based on the BT, a nonlocal symmetry and several families of exact solutions of this equation are obtained, including soliton solutions that have important physical significance. The Fitzhugh–Nagumo and Chaffee–Infante equations are also considered as special cases.

Keywords: “reaction–duffing” equation, Bäcklund transformation, symmetry, exact solution, soliton solution

DOI: https://doi.org/10.4213/tmf348

Full text: PDF file (180 kB)
References: PDF file   HTML file

English version:
Theoretical and Mathematical Physics, 2002, 132:1, 970–975

Bibliographic databases:

Revised: 21.01.2002

Citation: Yong Chen, Zhenya Yan, Hongqing Zhang, “Exact Solutions for a Family of Variable-Coefficient “Reaction–Duffing” Equations via the Bäcklund Transformation”, TMF, 132:1 (2002), 90–96; Theoret. and Math. Phys., 132:1 (2002), 970–975

Citation in format AMSBIB
\Bibitem{YonZheHon02} \by Yong Chen, Zhenya Yan, Hongqing Zhang \paper Exact Solutions for a~Family of Variable-Coefficient Reaction--Duffing'' Equations via the B\"acklund Transformation \jour TMF \yr 2002 \vol 132 \issue 1 \pages 90--96 \mathnet{http://mi.mathnet.ru/tmf348} \crossref{https://doi.org/10.4213/tmf348} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=1956679} \zmath{https://zbmath.org/?q=an:1129.35432} \transl \jour Theoret. and Math. Phys. \yr 2002 \vol 132 \issue 1 \pages 970--975 \crossref{https://doi.org/10.1023/A:1019663425564} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000177713500005} 

• http://mi.mathnet.ru/eng/tmf348
• https://doi.org/10.4213/tmf348
• http://mi.mathnet.ru/eng/tmf/v132/i1/p90

 SHARE:

Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. Chen, Y, “Soliton-like solutions for a (2+1)-dimensional nonintegrable KdV equation and a variable-coefficient KdV equation”, Nuovo Cimento Della Societa Italiana Di Fisica B-General Physics Relativity Astronomy and Mathematical Physics and Methods, 118:8 (2003), 767
2. Li, B, “Nonlinear partial differential equations solved by projective Riccati equations ansatz”, Zeitschrift fur Naturforschung Section A-A Journal of Physical Sciences, 58:9–10 (2003), 511
3. Chen, Y, “Generalized Riccati equation expansion method and its application to the Bogoyavlenskii's generalized breaking soliton equation”, Chinese Physics, 12:9 (2003), 940
4. Yong C, Qi W, “A series of new double periodic solutions to a (2+1)-dimensional asymmetric Nizhnik-Novikov-Veselov equation”, Chinese Physics, 13:11 (2004), 1796–1800
5. Chen Y, Wang Q, Li B, “Jacobi elliptic function rational expansion method with symbolic computation to construct new doubly-periodic solutions of nonlinear evolution equations”, Zeitschrift fur Naturforschung Section A-A Journal of Physical Sciences, 59:9 (2004), 529–536
6. Chen Y, Wang Q, Li B, “A generalized algebraic method for constructing a series of explicit exact solutions of a (1+1)-dimensional dispersive long wave equation”, Communications in Theoretical Physics, 42:3 (2004), 329–334
7. Wang Q, Chen Y, Li B, et al, “New exact travelling wave solutions to hirota equation and (1+1)-dimensional dispersive long wave equation”, Communications in Theoretical Physics, 41:6 (2004), 821–828
8. Chen Y, Wang Q, Li BA, “A generalized method and general form solutions to the Whitham-Broer-Kaup equation”, Chaos Solitons & Fractals, 22:3 (2004), 675–682
9. Chen Y, Li B, “New exact travelling wave solutions for generalized Zakharov-Kuzentsov equations using general projective Riccati equation method”, Communications in Theoretical Physics, 41:1 (2004), 1–6
10. Yomba E, “Construction of new soliton-like solutions of the (2+1) dimensional dispersive long wave equation”, Chaos Solitons & Fractals, 20:5 (2004), 1135–1139
11. Chen, Y, “General projective Riccati equation method and exact solutions for generalized KdV-type and KdV-Burgers-type equations with nonlinear terms of any order”, Chaos Solitons & Fractals, 19:4 (2004), 977
12. Chen Y, “General method and exact solutions to a generalized variable-coefficient two-dimensional KdV equation”, Nuovo Cimento Della Societa Italiana Di Fisica B-General Physics Relativity Astronomy and Mathematical Physics and Methods, 120:3 (2005), 295–302
13. Wang Q, Chen Y, Li B, et al, “New exact travelling wave solutions for the shallow long wave approximate equations”, Applied Mathematics and Computation, 160:1 (2005), 77–88
14. Chen Y, Wang Q, “A series of new soliton-like solutions and double-like periodic solutions of a (2+I)-dimensional dispersive long wave equation”, Chaos Solitons & Fractals, 23:3 (2005), 801–807
15. Wang Q, Chen Y, Zhang HQ, “A new Jacobi elliptic function rational expansion method and its application to (1+1)-dimensional dispersive long wave equation”, Chaos Solitons & Fractals, 23:2 (2005), 477–483
16. Chen, Y, “A new general algebraic method with symbolic computation to construct new doubly-periodic solutions of the (2+1)-dimensional dispersive long wave equation”, Applied Mathematics and Computation, 167:2 (2005), 919
17. Yu, YX, “The extended Jacobi elliptic function method to solve a generalized Hirota-Satsuma coupled KdV equations”, Chaos Solitons & Fractals, 26:5 (2005), 1415
18. Wang, Q, “An extended Jacobi elliptic function rational expansion method and its application to (2+1)-dimensional dispersive long wave equation”, Physics Letters A, 340:5–6 (2005), 411
19. Wang, Q, “A new Riccati equation rational expansion method and its application to (2+1)-dimensional Burgers equation”, Chaos Solitons & Fractals, 25:5 (2005), 1019
20. Chen, Y, “Elliptic equation rational expansion method and new exact travelling solutions for Whitham-Broer-Kaup equations”, Chaos Solitons & Fractals, 26:1 (2005), 231
21. Zeng, X, “Backlund transformation and exact solutions for (2+1) -dimensional Boussinesq equation”, Acta Physica Sinica, 54:4 (2005), 1476
22. Chen, Y, “A new Riccati equation rational expansion method and its application”, Zeitschrift fur Naturforschung Section A-A Journal of Physical Sciences, 60:1–2 (2005), 1
23. Chen, Y, “Extended Jacobi elliptic function rational expansion method and abundant families of Jacobi elliptic function solutions to (1+1)-dimensional dispersive long wave equation”, Chaos Solitons & Fractals, 24:3 (2005), 745
24. Li, B, “Soliton-like solutions and periodic form solutions for two variable-co efficient evolution equations using symbolic computation”, Acta Mechanica, 174:1–2 (2005), 77
25. Zeng, X, “New soliton-like solutions to the (2+1)-dimensional dispersive long wave equations”, Acta Physica Sinica, 54:2 (2005), 504
26. Zhang, XL, “A new generalized Riccati equation rational expansion method to generalized Burgers-Fisher equation with nonlinear terms of any order”, Communications in Theoretical Physics, 46:5 (2006), 779
27. Chen, Y, “A unified rational expansion method to construct a series of explicit exact solutions to nonlinear evolution equations”, Applied Mathematics and Computation, 177:1 (2006), 396
28. Wang, Q, “A multiple Riccati equations rational expansion method and novel solutions of the Broer-Kaup-Kupershmidt system”, Chaos Solitons & Fractals, 30:1 (2006), 197
29. Zeng, X, “Symbolic computation and new families of exact solutions to the (2+1)-dimensional dispersive long-wave equations”, Chaos Solitons & Fractals, 29:5 (2006), 1115
30. Chen, Y, “A new elliptic equation rational expansion method and its application to the shallow long wave approximate equations”, Applied Mathematics and Computation, 173:2 (2006), 1163
31. Li, B, “A generalized sub-equation expansion method and its application to the nonlinear Schrodinger equation in inhomogeneous optical fiber media”, International Journal of Modern Physics C, 18:7 (2007), 1187
32. Zhang, XL, “A new generalized Riccati equation rational expansion method to a class of nonlinear evolution equations with nonlinear terms of any order”, Applied Mathematics and Computation, 186:1 (2007), 705
33. Zheng, Y, “A new general algebraic method with symbolic computation to construct new exact analytical solution for a (2+1)-dimensional cubic nonlinear Schrodinger equation”, Chaos Solitons & Fractals, 32:3 (2007), 1101
34. Xiao Ya., Xue H., Zhang H., “An Extended Auxiliary Function Method and its Application in Mkdv Equation”, Math. Probl. Eng., 2013, 769187
35. Li Y., Zhao Yu., Yao Zh.-a., “Stochastic Exact Solutions of the Wick-Type Stochastic NLS Equation”, Appl. Math. Comput., 249 (2014), 209–221
36. Tian Shou-Fu, Ma Pan-Li, “On the Quasi-Periodic Wave Solutions and Asymptotic Analysis To a (3+1)-Dimensional Generalized Kadomtsev-Petviashvili Equation”, Commun. Theor. Phys., 62:2 (2014), 245–258
37. Kavitha L., Venkatesh M., Saravanan M., Dhamayanthi S., Gopi D., “Breather-Like Director Reorientations in a Nematic Liquid Crystal With Nonlocal Nonlinearity”, Wave Motion, 51:3 (2014), 476–488
38. Rui W., Zhao P., Zhang Yu., “Invariant Solutions and Conservation Laws of the (2+1)-Dimensional Boussinesq Equation”, Abstract Appl. Anal., 2014, 840405
39. Rui W., Zhang Yu., “Backlund Transformation and Quasi-Periodic Solutions For a Variable-Coefficient Integrable Equation”, Abstract Appl. Anal., 2014, 424059
40. Dai Ch.-Q., Xu Yu.-J., “Exact Solutions For a Wick-Type Stochastic Reaction Duffing Equation”, Appl. Math. Model., 39:23-24 (2015), 7420–7426
41. Xu M.-J., Tian Sh.-F., Tu J.-M., Ma P.-L., Zhang T.-T., “Quasi-Periodic Wave Solutions With Asymptotic Analysis To the Saweda-Kotera-Kadomtsev-Petviashvili Equation”, Eur. Phys. J. Plus, 130:8 (2015), 174
42. Xu M.-J., Tian Sh.-F., Tu J.-M., Zhang T.-T., “Bäcklund transformation, infinite conservation laws and periodic wave solutions to a generalized (2+1)-dimensional Boussinesq equation”, Nonlinear Anal.-Real World Appl., 31 (2016), 388–408
43. Choi J.H., Lee S., Kim H., “Stochastic Effects for the Reaction-Duffing Equation with Wick-Type Product”, Adv. Math. Phys., 2016, 9062343
44. Abdullah, Seadawy A.R., Jun W., “Mathematical Methods and Solitary Wave Solutions of Three-Dimensional Zakharov-Kuznetsov-Burgers Equation in Dusty Plasma and Its Applications”, Results Phys., 7 (2017), 4269–4277
45. Iqbal M., Seadawy A.R., Lu D., “Construction of Solitary Wave Solutions to the Nonlinear Modified Kortewege-de Vries Dynamical Equation in Unmagnetized Plasma Via Mathematical Methods”, Mod. Phys. Lett. A, 33:32 (2018), 1850183
46. Iqbal M., Seadawy A.R., Lu D., “Dispersive Solitary Wave Solutions of Nonlinear Further Modified Korteweg-de Vries Dynamical Equation in An Unmagnetized Dusty Plasma”, Mod. Phys. Lett. A, 33:37 (2018), 1850217
47. Abdullah, Seadawy A.R., Wang J., “Three-Dimensional Nonlinear Extended Zakharov-Kuznetsov Dynamical Equation in a Magnetized Dusty Plasma Via Acoustic Solitary Wave Solutions”, Braz. J. Phys., 49:1 (2019), 67–78
•  Number of views: This page: 518 Full text: 102 References: 19 First page: 1