Nothing Special   »   [go: up one dir, main page]
Skip to main content
Springer Nature Link
Log in

Analytical and numerical studies of the modified Kawahara equation with dual-power law nonlinearities

  • Original Paper
  • Published:
Numerical Algorithms Aims and scope Submit manuscript

Abstract

In this article, the Kawahara and modified Kawahara equations with dual-power law nonlinearities are solved based on the sine-cosine method and finite difference schemes. The fourth-order accurate difference scheme for the Kawahara equation and the second-order accurate difference scheme for the modified Kawahara equation have been constructed. The difference schemes have energy conservative properties. The existence, uniqueness, stability, and convergence of the numerical solutions are all well established. Finally, some numerical experiments demonstrate the reliability of the theoretical statements. Also, comparisons between the solutions obtained from the exact solitary wave solutions and the linearly finite difference schemes are made to demonstrate that the present schemes are efficient and reliable, and can simulate the single or multi-solitary waves propagating at a long period.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11

Similar content being viewed by others

Data availability

No data was used for the research described in the article.

References

  1. Darvishi, M.T., Khani, F.: Numerical and explicit solutions of the fifth-order Korteweg-de Vries equations. Chaos Soliton. Fract. 39, 2484–2490 (2009)

    Article  Google Scholar 

  2. Goswami, A., Singh, J., Kumar, D.: Numerical simulation of fifth-order KdV equations occurring in magneto-acoustic waves. Ain Shams Eng. J. 9, 2265–2273 (2018)

    Article  Google Scholar 

  3. Mei, L.Q., Chen, Y.P.: Numerical solutions of RLW equation using Galerkin method with extrapolation techniques. Comput. Phys. Commun. 183, 1609–1616 (2012)

    Article  MathSciNet  Google Scholar 

  4. Korteweg, D.J., de Vries, G.: On the change of form of long waves advancing in a rectangular canal, and on a new type of long stationary waves. Philos. Mag. 39, 422–443 (1895)

    Article  MathSciNet  Google Scholar 

  5. Wazwaz, A.M.: The extended tanh method for new solitons solutions for many forms of the fifth-order KdV equations. Appl. Math. Comput. 184, 1002–1014 (2007)

    MathSciNet  Google Scholar 

  6. Lee, C.T.: Some remarks on the fifth-order KdV equations. J. Math. Anal. Appl. 425, 281–294 (2015)

    Article  MathSciNet  Google Scholar 

  7. Ismail, H.N.A., Raslan, K.R., Salem, G.S.E.: Solitary wave solutions for the general KdV equation by Adomian decomposition method. Appl. Math. Comput. 154, 17–29 (2004)

    MathSciNet  Google Scholar 

  8. Bona, J.L., Dougalis, V.A., Karakashian, O.A., et al.: Conservative, high-order numerical schemes for the generalized Korteweg-de Vries equation. Philos. Trans. Royal Soc. London Ser. A 351, 107–164 (1995)

    Article  MathSciNet  Google Scholar 

  9. Başhan, A.: Bell-shaped soliton solutions and traveling wave solutions of the fifth-order nonlinear modified Kawahara equation. Int. J. Nonlin. Sci. Num. 22, 781–795 (2021)

    Article  MathSciNet  Google Scholar 

  10. Wazwaz, A.M.: Compactons and solitary patterns solutions to fifth-order KdV-like equations. Phys. A 371, 273–279 (2006)

    Article  Google Scholar 

  11. Salas, A.H.: Exact solutions for the general fifth KdV equation by the exp function method. Appl. Math. Comput. 205, 291–297 (2008)

    MathSciNet  Google Scholar 

  12. Saba, F., Jabeen, S., Akbar, H., et al.: Modified alternative (G’/G)-expansion method to general Sawada-Kotera equation of fifth-order. J. Egypt. Math. Soc. 23, 416–423 (2015)

    Article  MathSciNet  Google Scholar 

  13. Lax, P.D.: Integrals of nonlinear equations of evolution and solitary waves. Comm. Pure Appl. Math. 62, 467–490 (1968)

    Article  MathSciNet  Google Scholar 

  14. Wazwaz, A.M.: Multiple complex soliton solutions for the integrable KdV, fifth order Lax, modified KdV, Burgers, and Sharma-Tasso-Olver equations, Chinese. J. Phys. 59, 372–378 (2019)

    MathSciNet  Google Scholar 

  15. Jiang, X.H.: Asymptotic analysis of a perturbed periodic solution for KdV equation. Stud. Appl. Math. 116, 21–33 (2006)

    Article  MathSciNet  Google Scholar 

  16. Ito, M.: An extension of nonlinear evolution equations of the KdV (mKdV) type to higher orders. J. Phys. Soc. Jpn. 49, 771–778 (1980)

    Article  MathSciNet  Google Scholar 

  17. Wazwaz, A.M.: Solitons and periodic solutions for the fifth-order KdV equation. Appl. Math. Lett. 19, 1162–1167 (2006)

    Article  MathSciNet  Google Scholar 

  18. Al-Mdallal, Q.M., Syam, M.I.: Sine-cosine method for finding the soliton solutions of the generalized fifth-order nonlinear equation. Chaos Soliton. Fract. 33, 1610–1617 (2007)

    Article  MathSciNet  Google Scholar 

  19. Wongsaijai, B., Poochinapan, K.: A three-level average implicit finite difference scheme to solve equation obtained by coupling the Rosenau-KdV equation and the Rosenau-RLW equation. Appl. Math. Comput. 245, 289–304 (2014)

    MathSciNet  Google Scholar 

  20. Sukantamala, N., Nanta, S.: On solitary wave solutions for the Camassa-Holm and the Rosenau-RLW-Kawahara equations with the dual-power law nonlinearities. Abstract and Applied Analysis (2021). https://doi.org/10.1155/2021/6649285

  21. Mia, A.S., Akter, T.: Some exact solutions of two fifth-order KdV-type nonlinear partial differential equations. J. Partial Differ. Eqs. 25, 357–67 (2012)

    MathSciNet  Google Scholar 

  22. Wazwaz, A.M.: Multiple-soliton solutions for the fifth order Caudrey-Dodd-Gibbon (CDG) equation. Appl. Math. Comput. 197, 719–724 (2008)

    MathSciNet  Google Scholar 

  23. Kaya, D.: An explicit and numerical solutions of some fifth-order KdV equation by decomposition method. Appl. Math. Comput. 144, 353–363 (2003)

    MathSciNet  Google Scholar 

  24. Polat, N., Kaya, D., Tutalar, H.I.: A analytic and numerical solution to a modified Kawahara equation and a convergence analysis of the method. Appl. Math. Comput. 179, 466–472 (2006)

    MathSciNet  Google Scholar 

  25. Yuan, J.-M., Shen, J., Wu, J.: A dual-Petrov-Galerkin method for the Kawahara-type equations. J. Sci. Comput. 34, 48–63 (2008)

    Article  MathSciNet  Google Scholar 

  26. Khanal, N., Sharma, R., Wu, J., et al.: A dual-Petrov-Galerkin method for extended fifth-order Korteweg-de Vries type equations, Discrete Contin. Dyn. Syst. (2009) pp. 442–450

  27. Yuan, J.-M., Wu, J.: A dual-Petrov-Galerkin method for two integrable fifth-order KdV type equations. Discrete Contin. Dyn. Syst. 26, 1525–1536 (2010)

    Article  MathSciNet  Google Scholar 

  28. Ablowitz, M., Segur, H.: On the evolution of packets of water waves. J. Fluid Mech. 92, 691–715 (1979)

    Article  MathSciNet  Google Scholar 

  29. Ali, A., Kalisch, H.: On the formulation of mass, momentum and energy conservation in the KdV equation. Acta Appl. Math. 133, 113–131 (2014)

    Article  MathSciNet  Google Scholar 

  30. Nanta, S., Yimnet, S., Poochinapan, K., et al.: On the identification of nonlinear terms in the generalized Camassa-Holm equation involving dual-power law nonlinearities. Appl. Numer. Math. 160, 386–421 (2021)

    Article  MathSciNet  Google Scholar 

  31. He, D., Pan, K.: A linearly implicit conservative difference scheme for the generalized Rosenau-Kawahara-RLW equation. Appl. Math. Comput. 271, 323–336 (2015)

    MathSciNet  Google Scholar 

  32. Wazwaz, A.-M.: Partial Differential Equations and Solitary Waves Theory. Higher Education Press, Beijing (2009)

  33. Wang, X.: An energy-preserving finite difference scheme with fourth-order accuracy for the generalized Camassa-Holm equation. Commun. Nonlinear Sci. 119, 107121 (2023)

    Article  MathSciNet  Google Scholar 

  34. Ghiloufi, A., Omrani, K.: New conservative difference schemes with fourth-order accuracy for some model equations for nonlinear dispersive waves. Numer. Meth. Part. D. E. 34, 451–500 (2018)

    Article  MathSciNet  Google Scholar 

  35. Rouatbi, A., Omrani, K.: Two conservative difference schemes for a model of nonlinear dispersive equations. Chaos Soliton Fract. 104, 516–530 (2017)

    Article  MathSciNet  Google Scholar 

  36. Wang, X., Dai, W.: A three-level linear implicit conservative scheme for the Rosenau-KdV-RLW equation. J. Comput. Appl. Math. 330, 295–306 (2018)

    Article  MathSciNet  Google Scholar 

  37. Zhou, Y.: Application of Discrete Functional Analysis to the Finite Difference Methods. Inter. Acad. Publishers, Beijing (1990)

  38. Browder, F.E.: Existence and uniqueness theorems for solutions of nonlinear boundary value problems, Proceedings of Symposia in Applied Mathematics. Providence: AMS 17, 24–49 (1965)

  39. Morton, F.K., Mayers, D.F.: Numerical Solution of Partial Differential Equations, Cambridge University Press, (1994), Cambridge

  40. Wazwaz, A.M.: New soliton and periodic solutions for the fifth-order forms of the Lax and Sawada-Kotera equations. Int. J. Comput. Math. 84, 1663–1681 (2007)

    Article  MathSciNet  Google Scholar 

  41. Cesar, A.G.S.: Special forms of the fifth-order KdV equation with new periodic and soliton solutions. Appl. Math. Comput. 189, 1066–1077 (2007)

    MathSciNet  Google Scholar 

  42. Shi, F., Li, S., Xi, X.: High-order finite difference scheme for the fifth-order KdV equation. J. Hebei Normal University 41, 104–110 (2017)

    Google Scholar 

  43. Wang, X., Dai, W.: A new conservative finite difference scheme for the generalized Rosenau-KdV-RLW equation. Comput. Appl. Math. 39, 237 (2020)

    Article  MathSciNet  Google Scholar 

  44. Wongsaijai, B., Poochinapan, K., Disyadej, T.: A compact finite difference method for solving the General Rosenau-RLW equation. Int. J. Appl. Math. 44, 192–199 (2014)

    MathSciNet  Google Scholar 

  45. Wang, X., Dai, W.: A conservative fourth-order stable finite difference scheme for the generalized Rosenau-KdV equation in both 1D and 2D. J. Comput. Appl. Math. 55, 310–331 (2019)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

This work is supported by the Natural Science Foundation of Fujian Province, China (No., 2020J01796). The author also thanks the editors and reviewers for their constructive comments and suggestions which significantly improved the quality of this paper.

Author information

Authors and Affiliations

  1. School of Mathematics and Statistics, Minnan Normal University, Zhangzhou, Fujian, 363000, People’s Republic of China

    Xiaofeng Wang

Authors
  1. Xiaofeng Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

Contributions

The author declares he is the sole author of the manuscript.

Corresponding author

Correspondence to Xiaofeng Wang.

Ethics declarations

Competing interests

The author declares no competing interests.

Ethical approval

Written informed consent for the publication of this paper was obtained from Minnan Normal University.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Appendix

Appendix

In this appendix, we prove that the initial and boundary value problem (1.1)–(1.4) is well-posed.

Theorem A.1

Suppose \(u_{0}\in H_{0}^{2}[x_{l},x_{r}]\) and \(u(x,t)\in C_{x,t}^{5,1}([x_{l},x_{r}]\times [0,T])\), then the initial and boundary value problem (1.1)–(1.4) is well-posed.

Proof

First, we assume that \(u_{1}\) and \(u_{2}\) are two solutions of the problem (1.2)–(1.3) with the boundary condition (1.4) satisfying the initial conditions \(u_{0}^{(1)}\) and \(u_{0}^{(2)}\), respectively. Let \(\eta =u_{1}-u_{2}\), then \(\eta \) satisfies the following equation

$$\begin{aligned} \eta _{t}= & {} -\alpha (p+1)[(u_{1})^{p-1}(u_{1})_{x}(u_{1})_{xx}-(u_{2})^{p-1}(u_{2})_{x}(u_{2})_{xx}] -\alpha [(u_{1})^{p}(u_{1})_{xxx}\nonumber \\{} & {} -(u_{2})^{p}(u_{2})_{xxx}] -\beta [(u_{1})^{m}(u_{1})_{x}-(u_{2})^{m}(u_{2})_{x}]-\eta _{xxxxx}, \end{aligned}$$
(A.1)

with the initial condition

$$\begin{aligned} \eta (x,0)=u_{0}^{(1)}-u_{0}^{(2)},~~x\in [x_{l},x_{r}], \end{aligned}$$
(A.2)

and boundary conditions

$$\begin{aligned} \eta (x_{l},t)=\eta (x_{r},t)=0,~~\eta _{x}(x_{l},t)=\eta _{x}(x_{r},t)=0,~~\eta _{xx}(x_{l},t)=\eta _{xx}(x_{r},t)=0. \end{aligned}$$
(A.3)

Introducing the function \(\tilde{E}(t)=\displaystyle \int _{x_{l}}^{x_{r}}\eta ^{2}dx\) and by similar arguments as that in the proof of Theorem 2.1, we have

$$\begin{aligned} \frac{d\tilde{E}(t)}{d t}= & {} -2\alpha (p+1)\int _{x_{l}}^{x_{r}}\eta \Big [(u_{1})^{p-1}(u_{1})_{x}(u_{1})_{xx}-(u_{2})^{p-1}(u_{2})_{x}(u_{2})_{xx}\Big ]dx \nonumber \\{} & {} -2\int _{x_{l}}^{x_{r}}\eta \eta _{xxxxx}dx-2\alpha \int _{x_{l}}^{x_{r}}\eta \Big [(u_{1})^{p}(u_{1})_{xxx}-(u_{2})^{p}(u_{2})_{xxx}\Big ]dx \nonumber \\{} & {} -2\beta \int _{x_{l}}^{x_{r}}\eta \Big [(u_{1})^{m}(u_{1})_{x}-(u_{2})^{m}(u_{2})_{x}\Big ]dx. \end{aligned}$$
(A.4)

Since \(u(x,t)\in C_{x,t}^{5,1}([x_{l},x_{r}]\times [0,T])\), we let \(L=x_{l}-x_{r}\) and suppose that

$$\begin{aligned} M_{0}=\max \{|u|,|u_{x}|,|u_{xx}|,|u_{xxx}|\},~~(x,t)\in [x_{l},x_{r}]\times [0,T]. \end{aligned}$$

Thus, we have

$$\begin{aligned}{} & {} \Big |-2\alpha \int _{x_{l}}^{x_{r}}\eta \Big [(u_{1})^{p}(u_{1})_{xxx}-(u_{2})^{p}(u_{2})_{xxx}\Big ]dx\Big |\le 2\alpha M_{0}\Big |\int _{x_{l}}^{x_{r}}\eta \Big [(u_{1})^{p}-(u_{2})^{p}\Big ]dx\Big | \nonumber \\{} & {} =2\alpha M_{0}\Big |\int _{x_{l}}^{x_{r}}\eta (u_{1}-u_{2})\Big [(u_{1})^{p-1}+(u_{1})^{p-2}u_{2}+\cdot \cdot \cdot +u_{1}(u_{2})^{p-2}+(u_{2})^{p-1}\Big ]dx\Big | \nonumber \\{} & {} \le 2\alpha M_{0}pM^{p-1}_{0}\Big |\int _{x_{l}}^{x_{r}}\eta (u_{1}-u_{2})dx\Big |=2\alpha pM^{p}_{0}\int _{x_{l}}^{x_{r}}\eta ^{2}dx=2\alpha pM^{p}_{0}\tilde{E}(t). \end{aligned}$$
(A.5)

Similarly, we have

$$\begin{aligned}{} & {} \Big |-2\alpha (p+1)\int _{x_{l}}^{x_{r}}\eta \Big [(u_{1})^{p-1}(u_{1})_{x}(u_{1})_{xx}-(u_{2})^{p-1}(u_{2})_{x}(u_{2})_{xx}\Big ]dx\Big | \nonumber \\{} & {} \le 2\alpha (p+1)M_{0}^{2}\Big |\int _{x_{l}}^{x_{r}}\eta \Big [(u_{1})^{p-1}-(u_{2})^{p-1}\Big ]dx\Big | \nonumber \\{} & {} = 2\alpha (p+1)M_{0}^{2}\Big |\int _{x_{l}}^{x_{r}}\eta (u_{1}-u_{2})\Big [(u_{1})^{p-2}+(u_{1})^{p-3}u_{2}+\cdot \cdot \cdot +u_{1}(u_{2})^{p-3}\nonumber \\{} & {} \quad +(u_{2})^{p-2}\Big ]dx\Big | \le 2\alpha (p+1)M_{0}^{2}(p-1)M^{p-2}_{0}\Big |\int _{x_{l}}^{x_{r}}\eta (u_{1}-u_{2})dx\Big | \nonumber \\{} & {} =2\alpha (p^{2}-1)M^{p}_{0}\int _{x_{l}}^{x_{r}}\eta ^{2}dx=2\alpha (p^{2}-1)M^{p}_{0}\tilde{E}(t), \end{aligned}$$
(A.6)
$$\begin{aligned}{} & {} \Big |-2\beta \int _{x_{l}}^{x_{r}}\eta \Big [(u_{1})^{m}(u_{1})_{x}-(u_{2})^{m}(u_{2})_{x}\Big ]dx\Big | \nonumber \\{} & {} \le 2\beta M_{0}\Big |\int _{x_{l}}^{x_{r}}\eta \Big [(u_{1})^{m}-(u_{2})^{m}\Big ]dx\Big |\le 2\beta mM^{m}_{0}\int _{x_{l}}^{x_{r}}\eta ^{2}dx\nonumber \\{} & {} =2\beta mM^{m}_{0}\tilde{E}(t). \end{aligned}$$
(A.7)

Furthermore, we have

$$\begin{aligned} \int _{x_{l}}^{x_{r}}\eta \eta _{xxxxx}dx=\Big [(\eta \eta _{xxxx}-\eta _{x}\eta _{xxx})+\frac{1}{2}(\eta _{xx})^{2}\Big ]\Big |_{x_{l}}^{x_{r}}=0. \end{aligned}$$
(A.8)

Substituting (A.5)–(A.8) into (A.4), we obtain

$$\begin{aligned} \frac{d\tilde{E}(t)}{d t}\le \gamma E^{*}(t),~~t\in [0,T], \end{aligned}$$

which leads to

$$\begin{aligned} \tilde{E}(t)\le \tilde{E}(0)e^{\gamma t}\le \tilde{E}(0)e^{\gamma T}, \end{aligned}$$

where \(\gamma =2[\alpha (p^{2}+p-1)M^{p}_{0}+\beta mM^{m}_{0}]\). Thus, if \(u_{0}^{(1)}=u_{0}^{(2)}\), we have \(\eta (x,0)=0\) and hence \(\tilde{E}(0)=0\), implying that \(\tilde{E}(t)=0\), \(0\le t\le T\). By the Sobolev inequality [44, 45], we obtain \(\Vert \eta \Vert _{L_{2}}=0\) and \(u_{1}=u_{2}\). Furthermore, if \(\eta (x,0)<\varepsilon \), \(\eta _{x}(x,0)<\varepsilon \), \(\eta _{xx}(x,0)<\varepsilon \), we obtain \(\tilde{E}(0)<\varepsilon \) and hence \(\tilde{E}(t)\le \tilde{E}(0)e^{\gamma T}\le \varepsilon e^{\gamma T}\), where \(0\le t\le T\). That is, the solution is continuously dependent on the initial condition. Thus, the problems (1.1)–(1.4) are well-posed as required. \(\square \)

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Wang, X. Analytical and numerical studies of the modified Kawahara equation with dual-power law nonlinearities. Numer Algor (2024). https://doi.org/10.1007/s11075-024-01828-6

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s11075-024-01828-6

Keywords

Mathematics Subject Classification (2010)

Search

Navigation