Open Access
Issue
Math. Model. Nat. Phenom.
Volume 19, 2024
Article Number 9
Number of page(s) 13
Section Mathematical methods
DOI https://doi.org/10.1051/mmnp/2024004
Published online 18 June 2024
  1. D.J. Kordeweg and G. de Vries, XLI. 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 (1895) 422. [CrossRef] [MathSciNet] [Google Scholar]
  2. R.M. Miura, The Korteweg–de Vries equation: a survey of results. SIAM Rev. 18 (1976) 412. [CrossRef] [MathSciNet] [Google Scholar]
  3. A. Hasegawa and Y. Kodama, Solitons in Optical Communications. Oxford University Press, Oxford (1995). [Google Scholar]
  4. G.P. Agrawal, Nonlinear Fiber Optics. Academic Press, San Diego, CA (2007). [Google Scholar]
  5. J. Rubinstein, Sine-Gordon equation. J. Math. Phys. 11 (1970) 258. [CrossRef] [Google Scholar]
  6. F. Baronio, S. Wabnitz and Y. Kodama, Optical Kerr spatiotemporal dark-lump dynamics of hydrodynamic origin. Phys. Rev. Lett. 116 (2016) 173901. [CrossRef] [PubMed] [Google Scholar]
  7. H. Riaz and A. Wajahat, Multicomponent nonlinear Schrodinger equation in 2+1 dimensions, its Darboux transformation and soliton solutions. Eur. Phys. J. Plus 134 (2019) 222. [CrossRef] [Google Scholar]
  8. B.G. Konopelchenko and C. Rogers, On generalized Loewner systems: Novel integrable equations in 2+1 dimensions. J. Math. Phys. 34 (1993) 214. [CrossRef] [MathSciNet] [Google Scholar]
  9. A.S. Fokas, On the simplest integrable equation in 2+1. Inverse Probl. 10 (1994) L19. [CrossRef] [Google Scholar]
  10. G.W. Wang, K.T. Yang, H.C. Gu, F. Guan and A.H. Kara, A (2+1)-dimensional sine-Gordon and sinh-Gordon equations with symmetries and kink wave solutions. Nucl. Phys. B 953 (2020) 114956. [Google Scholar]
  11. H.C. Hu and X.D. Li, Nonlocal symmetry and interaction solutions for the new (3+1)-dimensional integrable Boussinesq equation. Math. Model. Nat. Phenom. 17 (2022) 2. [CrossRef] [EDP Sciences] [Google Scholar]
  12. S.Y. Lou, Deformations of the Riccati equation by using Miura-type transformations. J. Phys. A 30 (1997) 7259. [CrossRef] [MathSciNet] [Google Scholar]
  13. S.Y. Lou, Searching for higher dimensional integrable models from lower ones via Painlevé analysis. Phys. Rev. Lett. 80 (1998) 5027. [CrossRef] [MathSciNet] [Google Scholar]
  14. A.S. Fokas, Integrable nonlinear evolution partial differential equations in (4+2) and (3+1) dimensions. Phys. Rev. Lett. 96 (2006) 190201. [CrossRef] [MathSciNet] [PubMed] [Google Scholar]
  15. S.Y. Lou, X.Z. Hao and M. Jia, Deformation conjecture: deforming lower dimensional integrable systems to higher dimensional ones by using conservation laws. J. High Energy Phys. 2023 (2023) 018. [CrossRef] [Google Scholar]
  16. F.R. Wang and S.Y. Lou, Lax integrable higher dimensional Burgers systems via a deformation algorithm and conservation laws. Chaos Solitons Fract. 169 (2023) 113253. [CrossRef] [Google Scholar]
  17. M. Jia and S.Y. Lou, A novel (2+1)-dimensional nonlinear Schördinger equation deformed from (1+1)-dimensional nonlinear Schrödinger equation. Appl. Math. Lett. 143 (2023) 108684. [CrossRef] [Google Scholar]
  18. M. Casati and D.D. Zhang, Multidimensional integrable deformations of integrable PDEs. J. Phys. A: Math. Theor. 56 (2023) 505701. [CrossRef] [Google Scholar]
  19. G.H.F. Gardner, L.W. Gardner and A.R. Gregory, Formation velocity and density – the diagnostic basics for stratigraphic traps. Geophysics 39 (1974) 770. [CrossRef] [Google Scholar]
  20. R. Grimshaw, E. Pelinovsky and O. Poloukhina, Higher-order Korteweg–de Vries models for internal solitary waves in a stratified shear flow with a free surface. Nonlinear Process Geophys. 9 (2002) 221. [CrossRef] [Google Scholar]
  21. E.N. Pelinovskii, O.E. Polukhina and K. Lamb, Nonlinear internal waves in the ocean stratified in density and current. Oceanology 40 (2000) 757. [Google Scholar]
  22. A. Karczewska and P. Rozmej, (2+1)-dimensional KdV, fifth-order KdV, and Gardner equations derived from the ideal fluid model. Soliton, conidial and superposition solutions. Commun. Nonlinear Sci. Numer. Simul. 125 (2023) 107317. [CrossRef] [Google Scholar]
  23. Z.Z. Li, J.F. Han, D.N. Gao and W.S. Duan, Small amplitude double layers in an electronegative dusty plasma with distributed electrons. Chin. Phys. B 27 (2018) 105204. [CrossRef] [Google Scholar]
  24. C.K. Kuo, New solitary solutions of the Gardner equation and Whitham–Broer–Kaup equations by the modified simplest equation method. Optik 147 (2017) 128. [CrossRef] [Google Scholar]
  25. J.X. Fei, W.P. Cao and Z.Y. Ma, Nonlocal symmetries and explicit solutions for the Gardner equation. Appl. Math. Comput. 314 (2017) 293. [MathSciNet] [Google Scholar]
  26. H. Yepez-Martinez, M. Inc and H. Rezazadeh, New analytical solutions by the application of the modified double sub-equation method to the (1+1)-Schamel-KdV equation, the Gardner equation and the Burgers equation. Phys. Scr. 97 (2022) 085218. [CrossRef] [Google Scholar]
  27. K.J. Wang, Traveling wave solutions of the Gardner equation in dusty plasmas. Results Phys. 33 (2022) 105207. [CrossRef] [Google Scholar]
  28. G. Akram, S. Arshed, M. Sadaf, et al., Abundant solitary wave solutions of Gardner’s equation using three effective integration techniques. AIMS Math. 8 (2023) 8171. [CrossRef] [MathSciNet] [Google Scholar]

Current usage metrics show cumulative count of Article Views (full-text article views including HTML views, PDF and ePub downloads, according to the available data) and Abstracts Views on Vision4Press platform.

Data correspond to usage on the plateform after 2015. The current usage metrics is available 48-96 hours after online publication and is updated daily on week days.

Initial download of the metrics may take a while.