The Korteweg – de Vries – Burgers equation: with nonlinear source, reduction, the Painlevé test, first integrals and analytical solutions

The Korteweg-de Vries-Burgers equation with a nonlinear source is studied. The Cauchy problem for this equation cannot be solved by the inverse scattering transform in the general case. Therefore, the equation is considered taking into account the traveling wave variables. The Painlevé test is applied to the resulting nonlinear ordinary differential equation to investigate its integrability. It is shown that general solutions of the nonlinear ordinary differential equation are expressed via the Weierstrass elliptic function and the first Painlevé transcendents under certain parameter constraints. The relationship between the Painlevé test and special methods for finding exact solutions of nonlinear differential equations is discussed. Special methods are used to construct analytical solutions with one and two arbitrary constants. Exact solutions with two arbitrary constants expressed in terms of the Weierstrass elliptic function are obtained. Exact solutions with one arbitrary constant of the Korteweg-de Vries-Burgers equation with a nonlinear source are found using the logistic function method. It is demonstrated that the family of equations for which exact solutions are found is significantly expanded by the use of special methods.

1. Korteweg, D. J. de Vries, G. (1895) On the Change of Form of Long Waves advancing in a Rectangular Canal and on a New Type of Long Stationary Wave, Philosophical Magazine. 39 (240), 422-443, doi.org/10.1080/14786449508620739.

2. Russel J. S., (1844) Report on Waves, Rep. 14th Meet. Brit. Assoc> Adv. Sce., York, 1845, 311 – 390.

3. Zabusky, N.J. and Kruskal, M.D., (1965) Interaction of "solitons" in a collisionless plasma and the recurrence of initial states, Physical Review Letters, 15, 6, 240 – 243, doi = 10.1103/PhysRevLett.15.240.

4. Gardner, Clifford S. and Greene, John M. and Kruskal, Martin D. and Miura, Robert M., (1967) Method for solving the Korteweg-deVries equation, Physical Review Letters, 19, 1095 – 1097, doi = 10.1103/PhysRevLett.19.1095.

5. Kudryashov, N.A., (1988) Exact soliton solutions of the generalized evolution equation of wave dynamics, Journal of Applied Mathematics and Mechanics, 52, 3, 361 – 365, doi = 10.1016/0021-8928(88)90090-1.

6. Kudryashov, Nikolai A., (2009) On "new travelling wave solutions" of the KdV and the KdV-Burgers equations, Communications in Nonlinear Science and Numerical Simulation, 14, 5, 1891 – 1900, doi = 10.1016/j.cnsns.2008.09.020.

7. Feng, Zhaosheng and Wang, Xiaohui, (2003) The first integral method to the two-dimensional Burgers-Korteweg-de Vries equation, Physics Letters, Section A: General, Atomic and Solid State Physics, 308, 2 – 3, 173 – 178, doi = 10.1016/S0375-9601(03)00016-1.

8. Feng, Zhaosheng, (2002) The first-integral method to study the Burgers-Korteweg-de Vries equation, Journal of Physics A: Mathematical and General, 35, 2, 343 – 349, doi = 10.1088/0305-4470/35/2/312.

9. Bona, J.L., (1985) Travelling-wave solutions to the Korteweg-de Vries-Burgers equation, Proceedings of the Royal Society of Edinburgh: Section A Mathematics, 101, 3-4, 207 – 226, doi = 10.1017/S0308210500020783.

10. Parkes, E.J. and Duffy, B.R., (1997) Travelling solitary wave solutions to a compound KdV-Burgers equation, Physics Letters, Section A: General, Atomic and Solid State Physics, 229, 4, 217 – 220, doi = 10.1016/S0375-9601(97)00193-X.

11. El-Ajou, Ahmad and Arqub, Omar Abu and Momani, Shaher, (2015) Approximate analytical solution of the nonlinear fractional KdV-Burgers equation: A new iterative algorithm, Journal of Computational Physics, 293, 81 – 95, doi = 10.1016/j.jcp.2014.08.004.

12. Parkes, E.J. (1994) Exact solutions to the two-dimensional Korteweg-de Vries-Burgers equation, Journal of Physics A: Mathematical and General, 27, 13, L497 – L501, doi = 10.1088/0305-4470/27/13/006.

13. Soliman, A.A., A numerical simulation and explicit solutions of KdV-Burgers’ and Lax’s seventh-order KdV equations, Chaos, Solitons and Fractals, 29, 2, 294 – 302, doi = 10.1016/j.chaos.2005.08.054.

14. Feng, Zhaosheng, (2002) On explicit exact solutions to the compound Burgers-KdV equation, Physics Letters, Section A: General, Atomic and Solid State Physics, 293, 1-2, 57 – 66, doi = 10.1016/S0375-9601(01)00825-8.

15. Kudryashov, N.A., On types of nonlinear nonintegrable equations with exact solution, Physics Letters A, 1991, 155, 4-5, 269 – 275, doi=10.1016/0375-9601(91)90481-M.

16. Kudryashov, N.A., Logistic function as solution of many nonlinear differential equations, Applied Mathematical Modelling, 2015, 39, 18, 5733 – 5742, doi=10.1016/j.apm.2015.01.048.

17. Kudryashov, N.A., Painleve analysis and exact solutions of the Korteweg-de Vries equation with a source, Applied Mathematics Letters, 2015, 41, 41 – 45, doi=10.1016/j.aml.2014.10.015.

18. Fisher R.A., The wave of adance of advantageous genes, Ann. Eugenics, 1937, 7, 335 – 369.

19. McCue, Scott W. and El-Hachem, Maud and Simpson, Matthew J., Exact sharp-fronted travelling wave solutions of the Fisher-KPP equation, 2021, Applied Mathematics Letters, 114, doi = 10.1016/j.aml.2020.106918.

20. Lu, B.Q. and Xiu, B.Z. and Pang, Z.L. and Jiang, X.F., Exact traveling wave solution of one class of nonlinear diffusion equations, 1993, Physics Letters A, 175, 2, 113 – 115, doi = 10.1016/0375-9601(93)90131-I.

21. Kudryashov, N.A., Exact solutions of a family of Fisher equations, 1993, Theoretical and Mathematical Physics, 94, 2, 211 – 218, doi = 10.1007/BF01019332.

22. Aggarwal, S.K., Some numerical experiments on fisher equation, 1985, International Communications in Heat and Mass Transfer, 12, 4, 417 – 430, doi = 10.1016/0735-1933(85)90036-3.

23. Broadbridge, P. and Bradshaw-Hajek, B.H., Exact solutions for logistic reaction–diffusion equations in biology, year = 2016, Zeitschrift fur Angewandte Mathematik und Physik, 67, 4, doi = 10.1007/s00033-016-0686-3.

24. Kudryashov, Nikolai A., Exact solitary waves of the Fisher equation, 2005, Physics Letters, Section A: General, Atomic and Solid State Physics, 342, 1-2, 99 – 106, doi = 10.1016/j.physleta.2005.05.025.

25. Ebadi, Ghodrat and Biswas, Anjan, Application of the -expansion method for nonlinear diffusion equations with nonlinear source, 2010, Journal of the Franklin Institute, 347, 7, 1391 – 1398, doi = 10.1016/j.jfranklin.2010.05.013.

26. Hayek, Mohamed, Exact and traveling-wave solutions for convection-diffusion-reaction equation with power-law nonlinearity, 2011, Applied Mathematics and Computation, 218, 6, 2407 – 2420, doi = 10.1016/j.amc.2011.07.034.

27. Petrovskii, Sergei and Shigesada, Nanako, Some exact solutions of a generalized fisher equation related to the problem of biological invasion, 2001, Mathematical Biosciences, 172, 2, 73 – 94, doi = 10.1016/S0025-5564(01)00068-2.

28. Kowalevski, Sophie (1889), Sur le probleme de la rotation d’un corps solide autour d’un point fixe, Acta Mathematica, 12 (1): 177 – 232, doi:10.1007/BF02592182.

29. Kowalevski, Sophie (1890), Sur une propriété du systém d’ uations différentielles qui définit la rotation d’un corps solide autour d’un point fixe, Acta Mathematica, 14 (1): 81 – 93, doi:10.1007/BF02413316.

30. Ablowitz, M.J. and Ramani, A. and Segur, H., (1979),A connection between nonlinear evolution equations and ordinary differential equations of P-type. I, Journal of Mathematical Physics, 21, 4, 715 – 721, doi=10.1063/1.524491.

31. Ablowitz, M.J. and Ramani, A. and Segur, H., (1978), Nonlinear evolution equations and ordinary differential equations of painleve type, Lettere al Nuovo Cimento, 23, 9, 333-338, doi=10.1007/BF02824479.

32. Ablowitz, M.J. and Ramani, A. and Segur, H., (1979), A connection between nonlinear evolution equations and ordinary differential equations of P-type. II, Journal of Mathematical Physics, 21, 5, 1006 – 1015, doi=10.1063/1.524548.

33. Kudryashov, Nikolay A., Painleve analysis of the Sasa–Satsuma equation, 2024, Physics Letters, Section A: General, Atomic and Solid State Physics, 525, doi = 10.1016/j.physleta.2024.129900.

34. Kudryashov, Nikolay A., Painleve analysis of the resonant third-order nonlinear Schrödinger equation, 2024, Applied Mathematics Letters, 158, doi = 10.1016/j.aml.2024.109232.

35. Drazin P.G., Johnson R.S. Solitons: An Introduction. Cambridge: Cambridge University Press, 1989.

36. Ablowitz M.J., Clarkson P.A. Solitons Nonlinear Evolution Equations and Inverse Scattering. Cambridge: Cambridge University Press, 1991.

37. Kudryashov, Nikolay A., Mathematical model of propagation pulse in optical fiber with power nonlinearities, 2020, Optik, 212, doi = 10.1016/j.ijleo.2020.164750.

38. Kudryashov, Nikolai A., Simplest equation method to look for exact solutions of nonlinear differential equations, 2005, Chaos, Solitons and Fractals, 24, 5, 1217 – 1231, doi = 10.1016/j.chaos.2004.09.109.

39. Kudryashov, Nikolay A., One method for finding exact solutions of nonlinear differential equations, 2012, Communications in Nonlinear Science and Numerical Simulation, 17, 6, 2248 – 2253, doi = 10.1016/j.cnsns.2011.10.016.

40. Vitanov, Nikolay K. and Dimitrova, Zlatinka I. and Kantz, Holger, Modified method of simplest equation and its application to nonlinear PDEs, 2010, Applied Mathematics and Computation, 216, 9, 2587 – 2595, doi = 10.1016/j.amc.2010.03.102.

41. Ahmed, Hamdy M. and El-Sheikh, M.M.A. and Arnous, Ahmed H. and Rabie, Wafaa B., Construction of the Soliton Solutions for the Manakov System by Extended Simplest Equation Method, 2021, International Journal of Applied and Computational Mathematics, 7, 6, doi = 10.1007/s40819-021-01183-3.

42. Zayed, Elsayed M.E. and Shohib, Reham.M.A. and Al-Nowehy, Abdul-Ghani, On solving the (3+1)-dimensional NLEQZK equation and the (3+1)-dimensional NLmZK equation using the extended simplest equation method, 2019, Computers and Mathematics with Applications, 78, 10, 3390 – 3407, doi = 10.1016/j.camwa.2019.05.007.

43. Vitanov, Nikolay K., Modified method of simplest equation for obtaining exact solutions of nonlinear partial differential equations: History, recent developments of the methodology and studied classes of equations, 2019, Journal of Theoretical and Applied Mechanics (Bulgaria), 49, 2, 107 – 122.

44. Vitanov, Nikolay K. and Dimitrova, Zlatinka I., Modified method of simplest equation applied to the nonlinear Schrodinger equation, 2018, Journal of Theoretical and Applied Mechanics (Bulgaria), 48, 1, 59 – 68, doi = 10.2478/jtam-2018-0005.

45. Zayed, Elsayed M.E. and Shohib, Reham M.A., Optical solitons and other solutions to Biswas–Arshed equation using the extended simplest equation method, 2019, Optik, 185, 626 – 635, doi = 10.1016/j.ijleo.2019.03.112.

46. Chen, Cheng and Jiang, Yao-Lin, Simplest equation method for some time-fractional partial differential equations with conformable derivative, 2018, Computers and Mathematics with Applications, 75, 8, 2978 –- 2988, doi = 10.1016/j.camwa.2018.01.025.

47. Biswas, Anjan and Mirzazadeh, M. and Savescu, Michelle and Milovic, Daniela and Khan, Kaisar R. and Mahmood, Mohammad F. and Belic, Milivoj, Singular solitons in optical metamaterials by ansatz method and simplest equation approach, 2014, Journal of Modern Optics, 61, 19, 1550 – 1555, doi = 10.1080/09500340.2014.944357.

48. Vitanov, Nikolay K., On modified method of simplest equation for obtaining exact and approximate solutions of nonlinear PDEs: The role of the simplest equation, 2011, Communications in Nonlinear Science and Numerical Simulation, 16, 11, 4215 – 4231, doi = 10.1016/j.cnsns.2011.03.035.

49. Antonova, Anastasia O. and Kudryashov, Nikolay A., Generalization of the simplest equation method for nonlinear non-autonomous differential equations, 2014, Communications in Nonlinear Science and Numerical Simulation, 19, 11, 4037 – 4041, doi = 10.1016/j.cnsns.2014.03.035.

50. Zayed, Elsayed M.E. and Shohib, Reham M.A. and Al-Nowehy, Abdul-Ghani, Solitons and other solutions for higher-order NLS equation and quantum ZK equation using the extended simplest equation method, 2018, Computers and Mathematics with Applications, 76, 9, 2286 – 2303, doi = 10.1016/j.camwa.2018.08.027.

51. Khalique, Chaudry Masood, Solutions of a generalized complexly coupled Korteweg-de Vries system using simplest equation method, 2014, Proceedings – 2014 International Conference on Computational Science and Computational Intelligence, CSCI 2014, 2, 223 – 225, doi = 10.1109/CSCI.2014.123.

52. Kudryashov, Nikolai A. and Loguinova, Nadejda B., Extended simplest equation method for nonlinear differential equations, 2008, Applied Mathematics and Computation, 205, 1, 396 – 402, doi = 10.1016/j.amc.2008.08.019.

53. Seadawy, Aly R. and Yasmeen, Adeela and Raza, Nauman and Althobaiti, Saad, Novel solitary waves for fractional (2+1)-dimensional Heisenberg ferromagnetic model via new extendedgeneralized Kudryashov method, 2021, Physica Scripta, 96, 12, doi = 10.1088/1402-4896/ac30a4.

54. Zayed, Elsayed M. E. and El-Shater, Mona and Arnous, Ahmed H. and Yildirim, Yakup and Hussein, Layth and Jawad, Anwar Ja’afar Mohamad and Veni, S. Saravana and Biswas, Anjan, Quiescent optical solitons with Kudryashov’s generalized quintuple-power law and nonlocal nonlinearity having nonlinear chromatic dispersion with generalized temporal evolution by enhanced direct algebraic method and sub-ODE approach, 2024, European Physical Journal Plus, 139, 10, doi = 10.1140/epjp/s13360-024-05636-8.

55. Zhou, Jian and Ju, Long and Zhao, Shiyin and Zhang, Yufeng, Exact Solutions of Nonlinear Partial Differential Equations Using the Extended Kudryashov Method and Some Properties, 2023, Symmetry, 15, 12, doi = 10.3390/sym15122122.

56. Ekici, Mustafa, Exact Solutions to Some Nonlinear Time-Fractional Evolution Equations Using the Generalized Kudryashov Method in Mathematical Physics, 2023, Symmetry, 15, 10, doi = 10.3390/sym15101961.

57. Arnous, Ahmed H. and Biswas, Anjan and Yildirim, Yakup and Zhou, Qin and Liu, Wenjun and Alshomrani, Ali S. and Alshehri, Hashim M., Cubic–quartic optical soliton perturbation with complex Ginzburg–Landau equation by the enhanced Kudryashov’s method, 2022, Chaos, Solitons and Fractals, 155, doi = 10.1016/j.chaos.2021.111748.

58. Rabie, Wafaa B. and Ahmed, Hamdy M. and Hashemi, Mir Sajjad and Mirzazadeh, Mohammad and Bayram, Mustafa, Generating optical solitons in the extended (3 + 1)-dimensional nonlinear Kudryashov’s equation using the extended F-expansion method, 2024, Optical and Quantum Electronics, 56, 5, doi = 10.1007/s11082-024-06787-9.

59. Cinar, Melih and Secer, Aydin and Ozisik, Muslum and Bayram, Mustafa, Optical soliton solutions of (1 + 1)-and (2 + 1)-dimensional generalized Sasa-Satsuma equations using new Kudryashov method, 2023, International Journal of Geometric Methods in Modern Physics, 20, 2, doi = 10.1142/S0219887823500342.

60. Ali, Khalid K. and Mehanna, M.S. and Abdel-Aty, Abdel-Haleem and Wazwaz, Abdul-Majid, New soliton solutions of Dual mode Sawada Kotera equation using a new form of modified Kudryashov method and the finite difference method, 2024, Journal of Ocean Engineering and Science, 9, 3, 207 – 215, doi = 10.1016/j.joes.2022.04.033.

61. Ozisik, Muslum and Secer, Aydin and Bayram, Mustafa, On the investigation of chiral solitons via modified new Kudryashov method, 2023, International Journal of Geometric Methods in Modern Physics, 20, 7, doi = 10.1142/S0219887823501177.

62. Zayed, Elsayed M.E. and Gepreel, Khaled A. and Alngar, Mohamed E.M., Addendum to Kudryashov’s method for finding solitons in magneto-optics waveguides to cubic-quartic NLSE with Kudryashov’s sextic power law of refractive index, 2021, Optik, 230, doi = 10.1016/j.ijleo.2021.166311.

63. Rabie, Wafaa B. and Ahmed, Hamdy M., Construction cubic-quartic solitons in optical metamaterials for the perturbed twin-core couplers with Kudryashov’s sextic power law using extended F-expansion method, 2022, Chaos, Solitons and Fractals, 160, doi = 10.1016/j.chaos.2022.112289.