Araştırma Makalesi
Yıl 2024,
Cilt: 7 Sayı: 2, 75 - 81, 31.08.2024
Öz
Kaynakça
- [1] R.M. Miura, C.S. Gardner, M.D. Kruskal, Korteweg de Vries equation and generalizations. II. Existence of conservation laws and constants of motion, J. Math. Phys., 9 (1968), 1204-1209.
- [2] S. Watanabe, Ion acoustic soliton in plasma with negative ion, J. Phys. Soc. Japan, 53 (1984), 950-956.
- [3] M.S. Ruderman, T. Talipova, E. Pelinovsky, Dynamics of modulationally unstable ion-acoustic wavepackets in plasmas with negative ions, J. Plasma Phys., 74 (2008), 639-656.
- [4] R. Grimshaw, Environmental Stratified Flows, Topics in Environmental Fluid Mechanics, Kluwer, 2002.
- [5] E. Demler, A. Maltsev, Semiclassical solitons in strongly correlated systems of ultracold bosonic atoms in optical lattices, Ann. Phys., 326 (2011), 1775-1805.
- [6] A.H. Khater, A.A. Abdallah, O.H. El-Kalaawy, D.K. Callebaut, Backlund transformations, a simple transformation and exact solutions for dust-acoustic solitary waves in dusty plasma consisting of cold dust particles and two-temperature isothermal ions, Phys. Plasmas, 6 (1999), 4542-4547.
- [7] R. Grimshaw, D. Pelinovsky, E. Pelinovsky, T. Talipova, Wave group dynamics in weakly nonlinear long-wave models, Phys. D, 159 (2001), 35-37.
- [8] K.R. Helfrich, W.K. Melville, Long nonlinear internal waves, Annu. Rev. Fluid Mech., 38 (2006), 395-425.
- [9] J.R. Apel, L.A. Ostrovsky, Y.A. Stepanyants, J.F. Lynch, Internal solitons in the ocean and their effect on underwater sound, J. Acoust. Soc. Am., 121 (2007), 695-722
- [10] M. Wadati, Wave propagation in nonlinear lattice III, J. Phys. Soc. Jpn., 38 (1975), 681-686.
- [11] M. Coffey On series expansions giving closed form solutions of Korteweg de Vries like equations, J. Appl. Math., 50(6) (1990), 1580-1592.
- [12] S.Y. Lou, L.L Chen, Solitary wave solutions and cnoidal wave solutions to the combined KdV and mKdV equation, Math Meth. Appl. Sci., 17 (1994), 339-347.
- [13] J Zhang, New solitary wave solution of the combined KdV and mKdV equation, Int. Jour. Theo. Phys., 37 (1998), 1541-1546.
- [14] L. Lin, S. Zhu, Y. Xu, Y. Shi, Exact solutions of Gardner equations through tanh coth method, Appl. Math., 7 (2016), 2374-2381.
- [15] B. Ghanbari, D. Baleanu, New solutions of Gardner’s equation using two analytical methods, Front. In Phys., 7 (2015), 1-15.
- [16] M. Bokaeeyan, A. Ankiewicz, N. Akhmediev, Bright and dark rogue internal waves, the Gardner equation approach, Phys. Rev. E, 99 (2019), 062224-1-7.
- [17] A. Ankiewicz, M. Bokaeeyan, Integral relations for rogue wave formations of Gardner equation, Nonlinear Dyn, 99 (2020), 2939-2944.
- [18] P. Gaillard, The mKdV equation and multi-parameters rational solutions, Wave Motion, 100, (2021), 102667-1-9.
Yıl 2024,
Cilt: 7 Sayı: 2, 75 - 81, 31.08.2024
Öz
$N$-order solutions to the Gardner equation (G) are given in terms of Wronskians of order $N$ depending on $2N$ real parameters. We get solutions expressed with trigonometric or hyperbolic functions.
When one of the parameters goes to $0$, we succeed to get for each positive integer $N$, rational solutions as a quotient of polynomials in $x$ and $t$ depending on $2N$ real parameters. We construct explicit expressions of these rational solutions for the first orders.
Anahtar Kelimeler
Kaynakça
- [1] R.M. Miura, C.S. Gardner, M.D. Kruskal, Korteweg de Vries equation and generalizations. II. Existence of conservation laws and constants of motion, J. Math. Phys., 9 (1968), 1204-1209.
- [2] S. Watanabe, Ion acoustic soliton in plasma with negative ion, J. Phys. Soc. Japan, 53 (1984), 950-956.
- [3] M.S. Ruderman, T. Talipova, E. Pelinovsky, Dynamics of modulationally unstable ion-acoustic wavepackets in plasmas with negative ions, J. Plasma Phys., 74 (2008), 639-656.
- [4] R. Grimshaw, Environmental Stratified Flows, Topics in Environmental Fluid Mechanics, Kluwer, 2002.
- [5] E. Demler, A. Maltsev, Semiclassical solitons in strongly correlated systems of ultracold bosonic atoms in optical lattices, Ann. Phys., 326 (2011), 1775-1805.
- [6] A.H. Khater, A.A. Abdallah, O.H. El-Kalaawy, D.K. Callebaut, Backlund transformations, a simple transformation and exact solutions for dust-acoustic solitary waves in dusty plasma consisting of cold dust particles and two-temperature isothermal ions, Phys. Plasmas, 6 (1999), 4542-4547.
- [7] R. Grimshaw, D. Pelinovsky, E. Pelinovsky, T. Talipova, Wave group dynamics in weakly nonlinear long-wave models, Phys. D, 159 (2001), 35-37.
- [8] K.R. Helfrich, W.K. Melville, Long nonlinear internal waves, Annu. Rev. Fluid Mech., 38 (2006), 395-425.
- [9] J.R. Apel, L.A. Ostrovsky, Y.A. Stepanyants, J.F. Lynch, Internal solitons in the ocean and their effect on underwater sound, J. Acoust. Soc. Am., 121 (2007), 695-722
- [10] M. Wadati, Wave propagation in nonlinear lattice III, J. Phys. Soc. Jpn., 38 (1975), 681-686.
- [11] M. Coffey On series expansions giving closed form solutions of Korteweg de Vries like equations, J. Appl. Math., 50(6) (1990), 1580-1592.
- [12] S.Y. Lou, L.L Chen, Solitary wave solutions and cnoidal wave solutions to the combined KdV and mKdV equation, Math Meth. Appl. Sci., 17 (1994), 339-347.
- [13] J Zhang, New solitary wave solution of the combined KdV and mKdV equation, Int. Jour. Theo. Phys., 37 (1998), 1541-1546.
- [14] L. Lin, S. Zhu, Y. Xu, Y. Shi, Exact solutions of Gardner equations through tanh coth method, Appl. Math., 7 (2016), 2374-2381.
- [15] B. Ghanbari, D. Baleanu, New solutions of Gardner’s equation using two analytical methods, Front. In Phys., 7 (2015), 1-15.
- [16] M. Bokaeeyan, A. Ankiewicz, N. Akhmediev, Bright and dark rogue internal waves, the Gardner equation approach, Phys. Rev. E, 99 (2019), 062224-1-7.
- [17] A. Ankiewicz, M. Bokaeeyan, Integral relations for rogue wave formations of Gardner equation, Nonlinear Dyn, 99 (2020), 2939-2944.
- [18] P. Gaillard, The mKdV equation and multi-parameters rational solutions, Wave Motion, 100, (2021), 102667-1-9.
Toplam 18 adet kaynakça vardır.
Ayrıntılar
| Birincil Dil | İngilizce |
|---|---|
| Konular | Kısmi Diferansiyel Denklemler |
| Bölüm | Makaleler |
| Yazarlar | |
| Erken Görünüm Tarihi | 16 Temmuz 2024 |
| Yayımlanma Tarihi | 31 Ağustos 2024 |
| Gönderilme Tarihi | 27 Ocak 2024 |
| Kabul Tarihi | 23 Haziran 2024 |
| Yayımlandığı Sayı | Yıl 2024 Cilt: 7 Sayı: 2 |
Kaynak Göster
Journal of Mathematical Sciences and Modelling
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