×

A new ZK-BO equation for three-dimensional algebraic Rossby solitary waves and its solution as well as fission property. (English) Zbl 1390.76044

Summary: The BO equation depicts the evolution of algebraic Rossby solitary waves in ocean and atmosphere. But, despite its overt fame, the BO equation is restricted as a two-dimensional model. On the basis of the great success in the soliton theory, a lot of works have recently been directed to three-dimensional models and investigations of solitary waves properties in three-dimensional systems. In this paper, a new ZK-BO equation for three-dimensional algebraic Rossby solitary waves is derived by employing perturbation expansions and stretching transformations of time and space. By virtue of the trial function method, the exact solution of the ZK-BO equation is presented. By comparing the solution with the solution of two-dimensional algebraic Rossby solitary waves, we can find that the wave length of three-dimensional algebraic Rossby solitary waves is shorter while the wave amplitude is higher. Based on the exact solution of ZK-BO equation, the dissipation effect is studied. The results reveal the influence of dissipation effect on the three-dimensional algebraic solitary waves. Further, after theoretical analysis, the conservation laws of three-dimensional algebraic Rossby solitary waves are discussed and the fission property is studied.

MSC:

76B25 Solitary waves for incompressible inviscid fluids
35C08 Soliton solutions
37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.)
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Long, RR, Solitary waves in the westerlies, J. Atmos. Sci., 21, 197, (1964)
[2] Redekopp, LG, On the theory of solitary Rossby waves, J. Fluid. Mech., 82, 725, (1977) · Zbl 0362.76055
[3] Yang, HW; Yang, DZ; Shi, YL; Jin, SS; Yin, BS, Interaction of algebraic Rossby solitary waves with topography and atmospheric blocking, Dyn. Atmos. Oceans, 71, 21-34, (2015)
[4] Wadati, M, The modified Korteweg-de Vries equation, J. Phys. Soc. Jpn., 34, 1289, (1973) · Zbl 1334.35299
[5] Shi, Y.L., Yin, B.S., Yang, H.W., Yang, D.Z., Xu, Z.H.: Dissipative nonlinear Schrödinger equation for envelope solitary Rossby waves with dissipation effect in stratified fluids and its solution. Abs. Appl. Anal. 643652 (2014) · Zbl 1474.35589
[6] Boyd, JP, Equatorial solitary waves. part 1: Rossby solitons, J. Phys. Ocean., 10, 1699, (1980)
[7] Ono, H, Algebraic Rossby wave soliton, J. Phys. Soc. Jpn., 50, 2757, (1981)
[8] Meng, L; Lv, KL, Influences of dissipation on interaction of solitary wave with localized topography, Chin. J. Comput. Phys., 19, 349, (2002)
[9] Luo, DH, A barotropic envelope Rossby soliton model for block-eddy interaction. part I: effect of topography, J. Atmos. Sci., 62, 5, (2005)
[10] Yang, H.W., Xu, Z.H., Yang, D. Z., Feng, X.R., Yin, B.S., Dong, H.H.: ZK-Burgers equation for three-dimensional Rossby solitary waves and its solutions as well as chirp effect. Adv. Diff. Equ. 167, 1-22 (2016) · Zbl 1419.35180
[11] Xu, XX, An integrable coupling hierarchy of the mkdv-integrable systems, its Hamiltonian structure and corresponding nonisospectral integrable hierarchy, Appl. Math. Comput., 216, 344-353, (2010) · Zbl 1188.37065
[12] Yang, HX; Du, J; Xu, XX; Cui, JP, Hamiltionian and super-Hamiltonian systems of a hierarchy of siliton equations, Appl. Math. Comput., 217, 1497-1508, (2010) · Zbl 1202.35205
[13] Dong, HH; Guo, BY; Yin, BS, Generalized fractional supertrace identity for Hamiltonian structure of NLS-mkdv hierarchy with self-consistent sources, Anal. Math. Phys., 6, 199-209, (2016) · Zbl 1339.37051
[14] Dong, HH; Zhao, K; Yang, HW; Li, YQ, Generalised (2+1)-dimensional super mkdv hierarchy for integrable systems in soliton theory, East Asian J. Appl. Math., 5, 256-272, (2015) · Zbl 1457.35054
[15] Zhao, QL; Wang, XZ, The integrable coupling system of a 3x3 discrete matrix spectral problem, Appl. Math. Comput., 216, 730-743, (2010) · Zbl 1190.37063
[16] Li, XY; Zhao, QL, A new integrable symplectic map by the binary nonlinearization to the super AKNS system, J. Geom. Phys., 121, 123-137, (2017) · Zbl 1375.35438
[17] Zhang, Y; Dong, HH; Zhang, XE; Yang, HW, Rational solutions and lump solutions to the generalized (3+1)-dimensional shallow water-like equation, Comput. Math. Appl., 73, 246-252, (2017) · Zbl 1368.35240
[18] Ma, WX, Lump solutions to the Kadomtsev-Petviashvili equation, Phys. Lett. A, 379, 1975-1978, (2015) · Zbl 1364.35337
[19] Yan, ZY, Periodic, solitary and rational wave solutions of the 3D extended quantum Zakharov-Kuznetsov equation in dense quantum plasmas, Phys. Lett. A, 373, 2432-2437, (2009) · Zbl 1231.76362
[20] Ma, WX; Li, CX; He, J, A second Wronskian formulation of the Boussinesq equation, Nonlin. Anal. Theor. Methods Appl., 70, 4245-4258, (2009) · Zbl 1159.37425
[21] Yang, LG; Da, CJ; Song, J; etal., Rossby waves with linear topography in barotropic fluids, Chin. J. Oceanol. Limnol., 26, 334, (2008)
[22] Gao, ST, Nonlinear Rossby wave induced by large-scale topography, Adv. Atmos. Sci., 5, 301, (1988)
[23] Parkes, EJ; Duffy, BR; Abbott, PC, The Jacobi elliptic-function method for finding per iodic-wave solut ions to nonlinear evolution quations, Phys. Lett. A., 295, 280, (2001) · Zbl 1052.35143
[24] 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. Chin. Ed., 49, 965, (2006)
[25] Hassan, AZ; Aladrous, E; Shapl, S, Exact solutions for a perturbed nonlinear schr\(\ddot{o}\)dinger equation by using B\(\ddot{a}\)cklund transformations, Nonlin. Dyn., 74, 1145, (2013) · Zbl 1284.35410
[26] Ma, WX; Strampp, W, Bilinear forms and Bäcklund transformations of the perturbation systems, Phys. Lett. A., 341, 441, (2005) · Zbl 1171.37332
[27] Zhao, QL; Li, XY; Liu, FS, Two integrable lattice hierarchies and their respective Darboux transformations, Appl. Math. Comput., 219, 5693-5705, (2013) · Zbl 1288.37023
[28] Ma, WX; Zhang, RG, Adjont symmetry constraints of muliti-component AKNS equations, Chin. Ann. Math., 349, 153-163, (2006)
[29] Cao, R; Zhang, J, Trial function method and exact solutions to the generalized nonlinear schr\(\ddot{o}\)dinger equation with time-dependent coefficient, Chin. Phys. B., 22, 100507, (2013)
[30] Lee, JH; Ma, WX, A transformed rational function method and exact solutions to the 3+1 dimensional Jimbo-Miwa equation, Chaos Soliton. Fract., 42, 1356-1363, (2009) · Zbl 1198.35231
[31] Ma, WX, A refined invariant subspace method and applications to evolution equations, Sci. Chin. Math., 55, 1778-1796, (2012) · Zbl 1263.37071
[32] Ma, WX; Zhang, Y; Tang, Y; Tu, J, Hirota bilinear equations with linear subspaces of solutions, Appl. Math. Comput., 218, 7174-7183, (2012) · Zbl 1245.35109
[33] Kuo, H.L.: Atmospheric Dynamics. Science Press, Jiangsu (1981)
[34] Tung, KK; Lindzen, AS, A theory of stationary long waves, part I: a simple theory of blocking, Mon. Weather Rev., 107, 711, (1979)
[35] Charney, JG; De-Vore, JG, Multiple flow equilibria in the atmosphere and blocking, J. Atmos. Sci., 35, 1205, (1979)
[36] Luo, DH, On the Benjamin-Ono equation and its generalization in the atmosphere, Sci. Chin. (Ser. B), 32, 1233, (1989)
[37] Ahmet, B; Adem, CC, New solitons and periodic solutions for nonlinear physical models in mathematical physics, Nonlin. Anal. Real World Appl., 11, 3275, (2010) · Zbl 1196.35178
[38] You, ZJ, Simple and explicit solution to the wave dispersion equation, Coast. Eng., 48, 133, (2003)
[39] Zhang, ZB, Coexistence and stability of solutions for a class of reaction-diffusion systems, Electron. J. Diff. Equ., 137, 1, (2005)
[40] Ma, WX; Zhu, ZN, Solving the (3+1)-dimensional generalized KP and BKP equations by the multiple exp-function algorithm, Appl. Math. Comput., 218, 11871, (2012) · Zbl 1280.35122
[41] Ma, WX; You, Y, Solving the Korteweg-de Vries equation by its bilinear form: Wronskian solution, Trans. Am. Math. Soc., 307, 1753, (2005) · Zbl 1062.37077
[42] Abdou, MA, New solitons and periodic wave solutions for nonlinear physical models, Nonlin. Dyn., 52, 129, (2008) · Zbl 1173.35697
[43] Mcwilliams, J, An application of equivalent modons to atmospheric blocking, Dyn. Atmos. Oceans., 5, 219, (1980)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.