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ZK-Burgers equation for three-dimensional Rossby solitary waves and its solutions as well as chirp effect. (English) Zbl 1419.35180

Summary: Two-dimensional Rossby solitary waves propagating in a line have attracted much attention in the past decade, whereas there is few research on three-dimensional Rossby solitary waves. But as is well known, three-dimensional Rossby solitary waves are more suitable for real ocean and atmosphere conditions. In this paper, using multiscale and perturbation expansion method, a new Zakharov-Kuznetsov (ZK)-Burgers equation is derived to describe three-dimensional Rossby solitary waves that propagate in a plane. By analyzing the equation we obtain the conservation laws of three-dimensional Rossby solitary waves. Based on the sine-cosine method, we give the classical solitary wave solutions of the ZK equation; on the other hand, by the Hirota method we also obtain the rational solutions, which are similar to the solutions of the Benjamin-Ono (BO) equation, the solutions of which can describe the algebraic solitary waves. The rational solutions of the ZK equations are worth of attention. Finally, with the help of the classical solitary wave solutions, similar to the fiber soliton communication, we discuss the dissipation and chirp effect of three-dimensional Rossby solitary waves.

MSC:

35Q53 KdV equations (Korteweg-de Vries equations)
35Q51 Soliton equations
35C08 Soliton solutions
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[1] Gu, CH: Theory and Application of Solitary Wave. Zhejiang Science and Technology Press, Hangzhou (1990)
[2] Hasegawa, A: Optical Solitons in Fibers. Springer, Berlin (2003)
[3] Lou, SY, Tang, XY: Nonlinear Mathematical and Physics Methods. Science Press, Beijing (2006)
[4] Infeld, E, Rowlands, G: Nonlinear Waves, Solitons and Chaos. Cambridge University Press, Cambridge (2000) · Zbl 0994.76001
[5] Xu, ZH, Yin, BS, Hou, YJ, Xu, YS: Variability of internal tides and near-inertial waves on the continental slope of the northwestern South China Sea. J. Geophys. Res., Oceans 118, 197 (2013)
[6] Le, KC, Nguyen, LTK: Amplitude modulation of water waves governed by Boussinesq’s equation. Nonlinear Dyn. 81, 659 (2015) · Zbl 1347.76009
[7] Gao, XY: Comment on ‘Solitons, Bäcklund transformation, and Lax pair for the \((2+1)(2+1)\)-dimensional Boiti-Leon-Pempinelli equation for the water waves’. J. Math. Phys. 51, 093519 (2015) · Zbl 1306.37074
[8] Gao, XY: Bäcklund transformation and shock-wave-type solutions for a generalized \((3+1)(3+1)\)-dimensional variable-coefficient B-type Kadomtsev-Petviashvili equation in fluid mechanics. Ocean Eng. 96, 245 (2015)
[9] Yang, HL, Song, JB, Yang, LG, Liu, YJ: A kind of extended Korteweg-de Vries equation and solitary wave solutions for interfacial waves in a two-fluid system. Chin. Phys. 16, 3589 (2007)
[10] Gao, XY: Variety of the cosmic plasmas: general variable-coefficient Korteweg-de Vries-Burgers equation with experimental/observational support. Europhys. Lett. 110, 15002 (2015)
[11] Sun, WR, Tian, B, Liu, DY, Xie, XY: Nonautonomous matter-wave solitons in a Bose-Einstein condensate with an external potential. J. Phys. Soc. Jpn. 84, 074003 (2015)
[12] Xie, XY, Tian, B, Sun, WR, Wang, M, Wang, YP: Solitary wave and multi-front wave collisions for the Bogoyavlenskii-Kadomtsev-Petviashili equation in physics, biology and electrical networks. Mod. Phys. Lett. B 29, 1550192 (2015)
[13] Long, RR: Solitary waves in the westerlies. J. Atmos. Sci. 21, 197 (1964)
[14] Wadati, M: The modified Korteweg-de Vries equation. J. Phys. Soc. Jpn. 34, 34 (1973) · Zbl 1334.35299
[15] Yang, HW, Yin, BS, Dong, HH, Ma, ZD: Generation of solitary Rossby waves by unstable topography. Commun. Theor. Phys. 57, 473 (2012) · Zbl 1247.76022
[16] Meng, L, Lv, KL: Nonlinear long-wave disturbances excited by localized forcing. Chin. J. Comput. Phys. 17, 259 (2000)
[17] Yang, HW, Yin, BS, Shi, YL: Forced dissipative Boussinesq equation for solitary waves excited by unstable topography. Nonlinear Dyn. 70, 1389 (2012)
[18] Grimshaw, RHJ, Zhu, Y: Oblique interactions between internal solitary waves. Stud. Appl. Math. 92, 249 (1994) · Zbl 0813.76091
[19] Ma, WX: Combined Wronskian solutions to the 2D Toda molecule equation. Phys. Lett. A 375, 3931 (2011) · Zbl 1254.37045
[20] Qiao, ZJ, Li, ST: A new integrable hierarchy, parametric solutions and traveling wave solutions. Math. Phys. Anal. Geom. 7, 289 (2004) · Zbl 1068.37052
[21] Abdou, MA: New solitons and periodic wave solutions for nonlinear physical models. Nonlinear Dyn. 52, 129 (2008) · Zbl 1173.35697
[22] Ma, WX, Zhang, Y, Tang, YN, Tu, JY: Hirota bilinear equations with linear subspaces of solutions. Appl. Math. Comput. 218, 7174 (2012) · Zbl 1245.35109
[23] Zhao, Q, Liu, SK: Application of Jacobi elliptic functions in the atmospheric and oceanic dynamics: studies on two-dimensional nonlinear Rossby waves. Chin. J. Geophys. 49, 965 (2006)
[24] Zedan, HA, Aladrous, E, Shapll, S: Exact solutions for a perturbed nonlinear Schrödinger equation by using Bäcklund transformations. Nonlinear Dyn. 74, 1145 (2013) · Zbl 1284.35410
[25] Ono, H: Algebraic Rossby wave soliton. J. Phys. Soc. Jpn. 50, 2757 (1981)
[26] Yang, HW, Wang, XR, Yin, BS: A kind of new algebraic Rossby solitary waves generated by periodic external source. Nonlinear Dyn. 74, 1725 (2014) · Zbl 1314.76019
[27] Zhang, YF, Ma, WX: A study on rational solutions to a KP-like equation. Z. Naturforsch. A 70(4), 263 (2015)
[28] Shi, CG, Zhao, BZ, Ma, WX: Exact rational solutions to a Boussinesq-like equation in \((1+1)(1+1)\)-dimensions. Appl. Math. Lett. 48, 170 (2015) · Zbl 1326.35064
[29] Koch, TL: Optical Fiber Telecommunications IIIA. Academic Press, Pittsburgh (1997)
[30] Khalique, M, Magalakwe, G: Combined sinh-cosh-Gordon equation: symmetry reductions, exact solutions and conservation laws. Quaest. Math. 37, 199 (2014) · Zbl 1397.35152
[31] Biswas, A: 1-Soliton solution of the generalized Zakharov-Kuznetsov modified equal width equation. Appl. Math. Lett. 22, 1775 (2009) · Zbl 1179.35260
[32] Biswas, A, Zerrad, E: 1-Soliton solution of the Zakharov-Kuznetsov equation with dual-power law nonlinearity. Commun. Nonlinear Sci. Numer. Simul. 14, 3574 (2009) · Zbl 1221.35312
[33] Biswas, A: 1-Soliton solution of the generalized Zakharov-Kuznetsov equation with nonlinear dispersion and time-dependent coefficients. Phys. Lett. A 373, 2931 (2009) · Zbl 1233.35170
[34] Biswas, A, Zerrad, E: Solitary wave solution of the Zakharov-Kuznetsov equation in plasmas with power law nonlinearity. Nonlinear Anal., Real World Appl. 11, 3272 (2010) · Zbl 1196.35179
[35] Krishnan, EV, Biswas, A: Solutions to the Zakharov-Kuznetsov equation with higher order nonlinearity by mapping and ansatz methods. Phys. Wave Phenom. 18, 256 (2010)
[36] Suarez, P, Biswas, A: Exact 1-soliton solution of the Zakharov equation in plasmas with power law nonlinearity. Appl. Math. Comput. 217, 7372 (2011) · Zbl 1213.65134
[37] Johnpillai, AG, Kara, AH, Biswas, A: Symmetry solutions and reductions of a class of generalized \((2+1)(2+1)\)-dimensional Zakharov-Kuznetsov equation. Int. J. Nonlinear Sci. Numer. Simul. 12, 45 (2011) · Zbl 1401.35270
[38] Morris, R, Kara, AH, Biswas, A: Soliton solution and conservation laws of the Zakharov equation in plasmas with power law nonlinearity. Nonlinear Anal., Model. Control 18, 153 (2013) · Zbl 1298.35164
[39] Wang, GW, Xu, TZ, Johnson, S, Biswas, A: Solitons and Lie group analysis to an extended quantum Zakharov-Kuznetsov equation. Astrophys. Space Sci. 349, 317 (2014)
[40] Güner, Ö, Bekir, A, Moraru, L, Biswas, A: Bright and dark soliton solutions of the generalized Zakharov-Kuznetsov-Benjamin-Bona-Mahoney nonlinear evolution equation. Proc. Rom. Acad., Ser. A 16, 422 (2015)
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