Skip to main content
SpringerLink
Log in

Soliton solutions of weakly bound states for higher-order Ito equation

  • Original Paper
  • Published:
Nonlinear Dynamics Aims and scope Submit manuscript

Abstract

By utilizing an ingenious limit method proposed in this paper, the soliton solutions of weakly bound state including the multiple-pole solutions and the degenerate solution of breather solutions can be derived from the N-soliton solutions for higher-order Ito equation. By improving the traditional limit method, the dark double-pole solution can be obtained. Furthermore, some general forms of the multiple-pole solutions including the triple-pole solutions, quadruple-pole solutions, penta-pole solutions and the degenerate solution of breather solutions are derived. In addition, some dynamic behaviors of multiple-pole solutions are also specifically proposed. This limit method can also be applied to other integrable systems.

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

Access this article

Log in via an institution

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

Similar content being viewed by others

Data availibility

Our manuscript has no associated data.

References

  1. Lakomy, K., Nath, R., Santos, L.: Soliton molecules in dipolar Bose–Einstein condensates. Phys. Rev. A 86, 013610 (2012)

    Article  Google Scholar 

  2. Loginov, A.Y.: Nontopological solitons in the model of the self-interacting complex vector field. Phys. Rev. D. 91, 105028 (2015)

    Article  MathSciNet  Google Scholar 

  3. Zhang, R., Sha, W.Y., Jiang, G.R., Wang, J.G.: Soliton-like thermal source forcing and singular response of atmosphere and oceans to it. Appl. Math. Mech. 24, 714–719 (2003)

    Article  MATH  Google Scholar 

  4. Lou, S.Y.: Soliton molecules and asymmetric solitons in three fifth order systems via velocity resonance. J. Phys. Commun. 4, 041002 (2020)

    Article  Google Scholar 

  5. Liu, X.M., Yao, X.K., Cui, Y.D.: Real-time observation of the buildup of soliton molecules. Phys. Rev. Lett. 121, 023905 (2018)

    Article  Google Scholar 

  6. Rohrmann, P., Hause, A., Mitschke, F.: Solitons beyond binary: possibility of fibre-optic transmission of two bits per clock period. Sci. Rep. 2, 866–869 (2012)

    Article  Google Scholar 

  7. Ermolov, A., Mak, K.F., Frosz, M.H., et al.: Supercontinuum generation in the vacuum ultraviolet through dispersive-wave and soliton-plasma interaction in noble-gas-filled hollow-core photonic crystal fiber. Phys. Rev. A 92, 033821 (2015)

    Article  Google Scholar 

  8. Wang, L.H., Porsezian, K., He, J.S.: Breather and Rogue wave solutions of a generalized nonlinear Schrödinger equation. Phys. Rev. E 87, 053202 (2013)

    Article  Google Scholar 

  9. Zhang, Z., Chen, J.C., Guo, Q.: Multiple-pole solutions and degeneration of breather solutions to the focusing nonlinear Schrödinger equation. Commun. Theor. Phys. 74, 045002 (2022)

    Article  Google Scholar 

  10. Pu, J.C., Li, J., Chen, Y.: Soliton, breather, and rogue wave solutions for solving the nonlinear Schrödinger equation using a deep learning method with physical constraints. Chin. Phys. B 30, 06020 (2021)

    Article  Google Scholar 

  11. Zhang, Z., Yang, X.Y., Li, W.T., Li, B.: Trajectory equation of a lump before and after collision with line, lump, and breather waves for (2+1)-dimensional Kadomtsev–Petviashvili equation. Chin. Phys. B 28, 110201 (2019)

    Article  Google Scholar 

  12. Ma, W.X.: Lump solutions to the Kadomtsev–Petviashvili equation. Phys. Lett. A 379, 1975–1978 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  13. Wang, M., Tian, B., Liu, S.H., Shan, W.R., Jiang, Y.: Soliton, multiple-lump and hybrid solutions of a (2+1)-dimensional generalized Hirota–Satsuma–Ito equation for the water waves. Eur. Phys. J. Plus 136, 635 (2021)

    Article  Google Scholar 

  14. Rao, J.G., He, J.S., Mihalache, D., Chen, Y.: Dynamics of lump-soliton solutions to the PT-symmetric nonlocal Fokas system. Wave Motion 101, 102685 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  15. Wang, X., Yang, B., Chen, Y., Yang, Y.: Higher-order rogue wave solutions of the Kundu–Eckhaus equation. Phys. Scr. 89, 095210 (2014)

    Article  Google Scholar 

  16. Zhang, X.E., Chen, Y., Tang, X.Y.: Rogue wave and a pair of resonance stripe solitons to KP equation. Comput. Math. with Appl. 76, 1938–1949 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  17. Zhang, Z., Qi, Z.Q., Li, B.: Fusion and fission phenomena for (2+1)-dimensional fifth-order KdV system. Appl. Math. Lett. 16, 107004 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  18. Li, W.T., Li, J.H., Li, B.: Soliton molecules, asymmetric solitons and some new types of hybrid solutions in (2+1)-dimensional Sawada–Kotera model. Mod. Phys. B. 34, 2050141 (2020)

    Article  MathSciNet  Google Scholar 

  19. Herink, G., Kurtz, F., Jalali, B., Solli, D.R., Ropers, C.: Real-time spectral interferometry probes the internal dynamics of femtosecond soliton molecules. Science 356, 50–54 (2017)

    Article  Google Scholar 

  20. Wazwaz, A.M.: The tanh method: solitons and periodic solutions for the Dodd–Bullough–Mikhailov and the Tzitzeica–Dodd–Bullough equations. Chaos Soliton Fract. 25, 55–63 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  21. Wazwaz, A.M.: The sine–cosine and the tanh methods: reliable tools for analytic treatment of nonlinear dispersive equations. Appl. Math. Comput. 173, 150–164 (2006)

    MathSciNet  MATH  Google Scholar 

  22. Weng, W.F., Yan, Z.Y.: Inverse scattering and N-triple-pole soliton and breather solutions of the focusing nonlinear Schrödinger hierarchy with nonzero boundary conditions. Phys. Lett. A 407, 127472 (2021)

    Article  MATH  Google Scholar 

  23. Zhao, X., Tian, B., Tian, H.Y., Yang, D.Y.: Bilinear Bäcklund transformation, Lax pair and interactions of nonlinear waves for a generalized (2+1)-dimensional nonlinear wave equation in nonlinear optics/fluid mechanics/plasma physics. Nonlinear Dyn. 103, 1785–1794 (2021)

    Article  Google Scholar 

  24. Zhang, R.F., Li, M.C.: Bilinear residual network method for solving the exactly explicit solutions of nonlinear evolution equations. Nonlinear Dyn. 108, 521–531 (2022)

    Article  Google Scholar 

  25. Zhang, R.F., Li, M.C., Albishari, M., Zheng, F.C., Lan, Z.Z.: Generalized lump solutions, classical lump solutions and rogue waves of the (2+1)-dimensional Caudrey–Dodd–Gibbon–Kotera–Sawada-like equation. Appl. Math. Comput. 403, 126201 (2021)

    MathSciNet  MATH  Google Scholar 

  26. Zhang, R.F., Li, M.C., Gan, J.Y., Li, Q., Lan, Z.Z.: Novel trial functions and rogue waves of generalized breaking soliton equation via bilinear neural network method. Chaos Soliton Fract. 154, 111692 (2022)

    Article  MathSciNet  Google Scholar 

  27. Zhang, R.F., Li, M.C., Yin, H.M.: Rogue wave solutions and the bright and dark solitons of the (3+1)-dimensional Jimbo–Miwa equation. Nonlinear Dyn. 103, 1071–1079 (2021)

    Article  Google Scholar 

  28. Zhang, R.F., Bilige, S.: Bilinear neural network method to obtain the exact analytical solutions of nonlinear partial differential equations and its application to p-gBKP equation. Nonlinear Dyn. 95, 3041–3048 (2019)

    Article  MATH  Google Scholar 

  29. Li, Q., Li, M.C., Gong, Z.Q., Tian, Y., Zhang, R.F.: Locating and protecting interdependent facilities to hedge against multiple non-cooperative limited choice attackers. Reliab. Eng. Syst. Saf. 223, 108440 (2022)

    Article  Google Scholar 

  30. Hirota, R.: Exact solution of the Korteweg–de Vries equation for multiple collisions of solitons. Phys. Rev. Lett. 27, 1192–1194 (1971)

    Article  MATH  Google Scholar 

  31. Satsuma, J., Ablowitz, M.J.: Two-dimensional lumps in nonlinear dispersive systems. J. Math. Phys. 20, 1496–1503 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  32. Ohta, Y., Yang, J.K.: Rogue waves in the Davey–Stewartson I equation. Phys. Rev. E 85, 036604 (2012)

    Article  Google Scholar 

  33. Lou, S.Y.: Localized excitations of the (2+1)-dimensional sine-Gordon system. J. Phys. A Math. Gen. 36, 3877–3892 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  34. Li, M., Yue, X.L., Xu, T.: Multi-pole solutions and their asymptotic analysis of the focusing Ablowitz–Ladik equation. Phys. Scr. 95, 055222 (2020)

    Article  Google Scholar 

  35. Xing, Q.X., Wu, Z.W., Mihalache, D., He, J.S.: Smooth positon solutions of the focusing modified Korteweg–de Vries equation. Nonlinear Dyn. 89, 2299–2310 (2017)

    Article  MathSciNet  Google Scholar 

  36. Zhang, Z., Yang, X.Y., Li, B.: Novel soliton molecules and breather-positon on zero background for the complex modified KdV equation. Nonlinear Dyn. 100, 1551–1557 (2020)

    Article  Google Scholar 

  37. Dubrovsky, V.G.: The construction of exact multiple pole solutions of some (2+1)-dimensional integrable nonlinear evolution equations via the \({\overline{\partial }}\)-dressing method. J. Phys. A. 32, 369–390 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  38. Bilman, D., Buckingham, R.: Large-order asymptotics for multiple-pole solitons of the focusing nonlinear Schrödinger equation. J. Nonlinear Sci. 29, 2185–2229 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  39. Liu, W., Zhang, Y.S., He, J.S.: Dynamics of the smooth positons of the complex modified KdV equation. Wave Random Complex 28, 203–214 (2018)

    Article  MathSciNet  Google Scholar 

  40. Wadati, M., Ohkuma, K.: Multiple-Pole Solutions of the Modified Korteweg–de Vries Equation. J. Phys. Soc. Jpn. 51, 2029–2035 (1982)

    Article  MathSciNet  Google Scholar 

  41. Ito, M.: An extension of nonlinear evolution of the K-dV (mK-dV) type to higher orders. J. Phys. Soc. Jpn. 49, 771–778 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  42. Li, P., Pan, Z.L.: New periodic solutions of Ito’s 5th-order mKdV equation and Ito’s 7th-order mKdV equation. Appl. Math. Ser. B. 19, 44–50 (2004)

  43. Hu, X.B.: A Backlund transformation and nonlinear superposition formula of a higher-order Ito equation. J. Phys. A Math. Gen. 26, 5895–5903 (1993)

    Article  MATH  Google Scholar 

  44. Zhao, H.Q.: Soliton solution of a multi-component higher-order Ito equation. Appl. Math. Lett. 26, 681–686 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  45. Chen, D.Y., Zhang, D.J., Deng, S.F.: The novel multi-soliton solutions of the MKdV-sine Gordon equations. J. Phys. Soc. Jpn. 71, 658–659 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  46. Matveev, V.B.: Positon–positon and soliton–positon collisions: KdV case. Phys. Lett. A 166, 209–212 (1992)

    Article  MathSciNet  Google Scholar 

  47. Xing, Q.X., Wu, Z.W., Mihalache, D., He, J.S.: Smooth positon solutions of the focusing modified Korteweg–de Vries equation. Nonlinear Dyn. 89, 2299–2310 (2017)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

This work is supported by General Research Program of Wenzhou Polytechnic (Nos. XJ2022000104), National Natural Science Foundation of China under Grant Nos. 12175111, and K.C.Wong Magna Fund in Ningbo University.

Funding

General Research Program of Wenzhou Polytechnic (Nos. XJ2022000104).

Author information

Authors and Affiliations

  1. School of General Education, Wenzhou Polytechnic, Wenzhou, 325000, People’s Republic of China

    Wen-Tao Li

  2. School of Mathematics and Statistics, Ningbo University, Ningbo, 315211, People’s Republic of China

    Biao Li

Authors
  1. Wen-Tao Li
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Biao Li
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Wen-Tao Li.

Ethics declarations

Conflict of interest

The authors declare that they have no conflict of interest.

Additional information

Publisher's Note

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

Appendix

Appendix

In Fig. 5, the mathematical expression of penta-pole solution is shown as follows:

$$\begin{aligned} u_{5-p}=-i\left[ \ln \frac{g}{f}\right] _{x}, \end{aligned}$$
(17)

where

$$\begin{aligned} f=g^{*}, \end{aligned}$$
(18)

with

$$\begin{aligned} g&= -243\,i{\mathrm{e}^{-5\,t+5\,x}}+ \left( 129654\,{t}^{4}-74088\,{t}^{3}x\right. \nonumber \\&\quad + 15876\,{t}^{2}{x}^{2}-1512\,t{x}^{3}+54\,{x}^{4}+889056\,{t}^{3} \nonumber \\&\quad -285768\,{t}^{2}x+27216\,t{x}^{2}-648\,{x}^{3}\nonumber \\&\quad +1309770\,{t}^{2}-165564 \,tx+2754\,{x}^{2}\nonumber \\&\quad \left. +328860\,t-4536\,x +2430 \right) {\mathrm{e}^{-4\,t+4\,x}}\nonumber \\&\quad + \left( 1210104\,i{t}^{5}x-432180\,i{t}^{4}{x}^{2}\right. \nonumber \\&\quad +82320\,i{t}^{3}{x}^{3}-8820\,i{t}^{2}{x}^{4}+504\,it{x}^{5}\nonumber \\&\quad +3803184\,i{t}^{4}x-938448\,i{t}^{3}{x}^{2}+112896\,i{t}^{2}{x}^{3} \nonumber \\&\quad -6552\,it{x}^{4}+4050144\,i{t }^{3}x-465696\,i{t}^{2}{x}^{2} \nonumber \\&\quad +24192\,it{x}^{3}+1629936\,i{t}^{2}x+3024\,it{x}^{2}\nonumber \\&\quad -176148\,itx-6050520\,i{t}^{5}-11928168\,i{t}^{4}\nonumber \\&\quad -4395888\,i{t}^{3}-3087882\,i{t}^{2}+276696\,it\nonumber \\&\quad -1485\,i-12\,i{x}^{6} -792\,i{x}^{4}-4266\,i{x}^{2}\nonumber \\&\quad \left. -1411788\,i{t}^{6} +144\,i{x}^{5}+2448\,i{x}^{3}+3780\,ix \right) \nonumber \\&\quad {\mathrm{e}^{-3\,t+3\,x}}\!+\! \left( 470596\,{t}^{6}\!-\! 403368\,{t}^{5}x\!+\!144060\,{t}^{4}{x}^{2}\right. \nonumber \\&\quad -27440\,{t}^{3}{x}^{3} +2940\,{t}^{2}{x}^{4}-168\,t{x}^{5}\nonumber \\&\quad +4\,{x}^{6}-806736\,{t}^{5}+403368\,{t}^{4}x -65856\,{t}^{3}{x}^{2}\nonumber \\&\quad +2352\,{t}^{2}{x}^{3}+336\,t{x}^{4} -24\,{x}^{5}+1483818\,{t}^{4}\nonumber \\&\quad -222264\,{t}^{3}x-22932\,{t}^{2}{x}^{2}+2856\,t{x}^{3} \nonumber \\&\quad +90\,{x}^{4}+452760\,{t}^{3}+324576\,{t}^{2}x +504\,t{x}^{2}\nonumber \\&\quad -240\,{x}^{3}-30870\,{t}^{2}-5292\,tx\nonumber \\&\quad \left. +450\,{x}^{2}+5040\,t-540\,x+315 \right) {\mathrm{e}^{-2\,t+2\,x}} \nonumber \\&\quad +1+\left( -4802\,i{t}^{4}+2744\,i{t}^{3}x\right. \nonumber \\&\quad -588\,i{t}^{2}{x}^{2}+56\,it{x}^{3}-2\,i{x}^{4}+24696\,i{t}^{3}\nonumber \\&\quad -7056\,i{t}^{2 }x+504\,it{x}^{2}-22344\,i{t}^{2}+1680\,itx\nonumber \\&\quad \left. +1680\,it \right) {\mathrm{e}^{-t+x}}. \end{aligned}$$
(19)

Here, \(*\) denotes the conjugate.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Li, WT., Li, B. Soliton solutions of weakly bound states for higher-order Ito equation. Nonlinear Dyn 110, 741–751 (2022). https://doi.org/10.1007/s11071-022-07662-6

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11071-022-07662-6

Keywords

Search

Navigation