×

Combined ZK-mzk equation for Rossby solitary waves with complete Coriolis force and its conservation laws as well as exact solutions. (English) Zbl 1445.35280

Summary: In the present paper, a new partial differential equation has been obtained to describe the Rossby solitary waves with complete Coriolis force by employing multi-scale analysis and perturbation method, we call it combined ZK-mZK equation. The equation can reflect the propagation of Rossby waves on the plane and is more appropriate for the real ocean and atmosphere than the \((1+1)\) dimensional models (such as KdV and mKdV), which can only represent the propagation of Rossby solitary waves in a line. Furthermore, by adopting the multiplier method, we construct conservation laws of the combined ZK-mZK equation, which is meaningful for researching the global stability of solutions. Finally, we deduce the exact solutions of the combined ZK-mZK equation via the semi-inverse variational principle. By applying these exact solutions, some propagation features of Rossby solitary waves are analyzed.

MSC:

35Q51 Soliton equations
35C08 Soliton solutions
37K40 Soliton theory, asymptotic behavior of solutions of infinite-dimensional Hamiltonian systems
37K45 Stability problems for infinite-dimensional Hamiltonian and Lagrangian systems
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Ambrizzi, T, Hoskins, BJ, Hsu, HH: Rossby wave propagation and teleconnection patterns in the austral winter. J. Atmos. Sci. 52, 3661 (1970)
[2] Grimshaw, R: Nonlinear aspects of long shelf waves. Geophys. Astrophys. Fluid Dyn. 8, 3 (1977) · Zbl 0353.76073
[3] Redekopp, LG: On the theory of solitary Rossby waves. J. Fluid Mech. 82, 725 (1977) · Zbl 0362.76055
[4] Wadati, M: The modified Korteweg-deVries equation. J. Phys. Soc. Jpn. 34, 1289 (1973) · Zbl 1334.35299
[5] Chen, JC, Zhu, SD: Residual symmetries and soliton-cnoidal wave interaction solutions for the negative-order Korteweg-de Vries equation. Appl. Math. Lett. 73, 136 (2017) · Zbl 1375.35018
[6] Song, J, Yang, LG: Modified KdV equation for solitary Rossby waves with effect in barotropic fluids. Chin. Phys. B 7, 2873 (2009)
[7] Xu, XX: An integrable coupling hierarchy of the MKdv-integrable systems, its Hamiltonian structure and corresponding nonisospectral integrable hierarchy. Appl. Math. Comput. 216, 344 (2010) · Zbl 1188.37065
[8] Biswas, A: Topological and non-topological solitons for the generalized Zakharov-Kuznetsov modified equal width equation. Int. J. Theor. Phys. 48, 2698 (2009) · Zbl 1187.81098
[9] Zhang, RG, Yang, LG, Song, J, Yang, HL: \((2+1)(2+1)\) dimensional Rossby waves with complete Coriolis force and its solution by homotopy perturbation method. Comput. Math. Appl. 73, 1996 (2017) · Zbl 1371.86021
[10] Kumar, D, Singh, J, Baleanu, D: Modified Kawahara equation within a fractional derivative with non-singular kernel. Therm. Sci. (2017). https://doi.org/10.2298/TSCI160826008K
[11] Yang, HW, Xu, ZH, Yang, DZ, Feng, XR, Yin, BS, Dong, HH: ZK-Burgers equation for three-dimensional Rossby solitary waves and its solutions as well as chirp effect. Adv. Differ. Equ. 2016, 167 (2016) · Zbl 1419.35180
[12] Yang, HW, Zhao, QF, Dong, HH: A new integrodifferential equation for Rossby solitary waves with topography effect in deep rotational fluids. Abstr. Appl. Anal. 2013, 597807 (2013)
[13] Yang, HW, Yin, BS, Shi, YL, Wang, QB: Forced ILW-Burgers equation as a model for Rossby solitary waves generated by topography in finite depth fluids. J. Appl. Math. 2012, 491343 (2012) · Zbl 1267.35080
[14] Yang, JY, Ma, WX, Qin, ZY: Lump and lump-soliton solutions to the \((2+1)(2+1)\)-dimensional Ito equation. Anal. Math. Phys. (2017). https://doi.org/10.1007/s13324-017-0181-9 · Zbl 1403.35261
[15] Philips, NA: Reply to G. Veronis’s comments on Phillips (1966). J. Atmos. Sci. 25, 115 (1968)
[16] Veronis, G: Comments on Phillips’s (1966) proposed simplification of the equations of motion for shallow rotating atmosphere. J. Atmos. Sci. 5, 1154 (1968)
[17] Wangsness, RK: Comments on “The equations of motion for a shallow rotating atmosphere and the ‘traditional approximation’’’. J. Atmos. Sci. <Emphasis Type=”Bold”>27, 504 (1970)
[18] White, AA, Bromley, RA: Dynamically consistent, quasi-hydrostatic equations for global models with a complete representation of the Coriolis force. Q. J. R. Meteorol. Soc. 121, 399-418 (1995)
[19] Gerkema, T, Shrira, VI: Near-inertial waves on the “nontraditional” β-plane. J. Geophys. Res. 110, C01003 (2005)
[20] Dellar, PJ, Salmon, R: Shallow water equations with a complete Coriolis force and topography. Phys. Fluids 17, 106601 (2005) · Zbl 1188.76039
[21] Dellar, PJ: Variations on a β-plane: derivation of non-traditional β-plane equations from Hamilton’s principle on a sphere. J. Fluid Mech. 674, 174 (2011) · Zbl 1241.76428
[22] Ma, HJ, Jia, YM: Stability analysis for stochastic differential equations with infinite Markovian switchings. J. Math. Anal. Appl. 435, 593 (2016) · Zbl 1323.93071
[23] Tramontana, F, Elsadany, AA, Xin, BG, Agiza, HN: Local stability of the Cournot solution with increasing heterogeneous competitors. Nonlinear Anal., Real World Appl. 26, 150 (2015) · Zbl 1329.91095
[24] Yan, ZG, Zhang, WH: Finite-time stability and stabilization of Ito-type stochastic singular systems. Abstr. Appl. Anal. 2014, 263045 (2014)
[25] Yang, HL, Liu, FM, Wang, DN: Nonlinear Rossby waves near the equator with complete Coriolis force. Prog. Geophys. 31, 0988 (2016)
[26] Hayashi, M, Itoh, H: The importance of the nontraditional Coriolis terms in large-scale motions in the tropics forced by prescribed cumulus heating. J. Atmos. Sci. 69, 2699 (2012)
[27] Madden, RA, Julian, PR: Detection of a 40-50 day oscillation in the zonal wind in the tropical Pacific. J. Atmos. Sci. 28, 702 (1971)
[28] Miura, H, Satoh, M, Nasuno, T: A Madden-Julian oscillation event realistically simulated by a global cloud-resolving model. Science 318, 1763 (2007)
[29] Madden, RA, Julian, PR: Description of global-scale circulation cells in the tropics with a 40-50 day period. J. Atmos. Sci. 29, 1109 (1972)
[30] Maloney, ED, Sobel, AH, Hannah, WM: Intraseasonal variability in an aquaplanet general circulation model. J. Adv. Model. Earth Syst. 2, 5 (2010)
[31] Landu, K, Maloney, ED: Effect of SST distribution and radiative feedbacks on the simulation of intra seasonal variability in an aqua planet GCM. J. Meteorol. Soc. Jpn. 89, 195 (2011)
[32] Tang, LY, Fan, JC: A family of Liouville integrable lattice equations and its conservation laws. Appl. Math. Comput. 217, 1907 (2010) · Zbl 1202.39005
[33] Li, XY, Zhang, YQ, Zhao, QL: Positive and negative integrable hierarchies, associated conservation laws and Darboux transformation. J. Comput. Appl. Math. 233, 1096 (2009) · Zbl 1184.37055
[34] Laplace, PS: Trait de Mcanique Cleste, Celestial Mechanics New York (1966)
[35] Steudel, H: Uber die Zuordnung zwischen invarianzeigen-schaften and Erhaltungssatzen. Z. Naturforsch. 17A, 129 (1962)
[36] Kumar, D, Singh, J, Baleanu, D: A new analysis for fractional model of regularized long-wave equation arising in ion acoustic plasma waves. Math. Methods Appl. Sci. 40, 5642 (2017) · Zbl 1388.35212
[37] Anco, SC, Bluman, GW: Direct construction method for conservation laws of partial differential equations. Part I: examples of conservation law classifications. Eur. J. Appl. Math. 13, 545 (2002) · Zbl 1034.35070
[38] Biswas, A, Kara, AH, Bokhari, AH, Zaman, FD: Solitons and conservation laws of Klein Gordon equation with power law and log law nonlinearities. Nonlinear Dyn. 73, 2191 (2013) · Zbl 1281.35069
[39] Liu, XH, Zhang, WG, Li, ZM: The orbital stability of the solitary wave solutions of the generalized Camassa-Holm equation. J. Math. Anal. Appl. 398, 776 (2013) · Zbl 1253.35139
[40] Ibragimov, NH: A new conservation theorem. J. Math. Anal. Appl. 333, 311 (2007) · Zbl 1160.35008
[41] Ibragimov, NH: Integrating factors, adjoint equations and Lagrangians. J. Math. Anal. Appl. 318, 742 (2006) · Zbl 1102.34002
[42] Ibragimov, NH, Shabat, AB: Korteweg-de Vries equation from the group-theoretic point of view. Sov. Phys. Dokl. 24, 15 (1979) · Zbl 0423.35076
[43] Wazwaz, AM: Compact and noncompact physical structures for the ZK-BBM equation. Appl. Math. Comput. 169, 713 (2005) · Zbl 1078.35527
[44] Wang, ML: Solitary wave solution for variant Boussinesq equations. Phys. Lett. A 199, 169 (1995) · Zbl 1020.35528
[45] Ma, WX: Trigonal curves and algebro-geometric solutions to soliton hierarchies I. Proc. R. Soc. A 473, 20170232 (2017) · Zbl 1404.35392
[46] Ma, WX: Trigonal curves and algebro-geometric solutions to soliton hierarchies II. Proc., Math. Phys. Eng. Sci. 473, 20170233 (2017) · Zbl 1404.35393
[47] Khalfallah, M: Exact traveling wave solutions of the Boussinesq-Burgers equation. Math. Comput. Model. 49, 666 (2009) · Zbl 1165.35445
[48] Li, XY, Zhao, QL, Li, YX, Dong, HH: Binary Bargmann symmetry constraint associated with 3×\(33\times 3\) discrete matrix spectral problem. J. Nonlinear Sci. Appl. 8, 496 (2015) · Zbl 1327.35335
[49] Tian, ZL, Tian, MY, Liu, ZY, Xu, TY: The Jacobi and Gauss-Seidel-type iteration methods for the matrix equation AXB = C. Appl. Math. Comput. 292, 63 (2017)
[50] Kumar, D, Singh, J, Kumar, S, Sushila: Numerical computation of Klein-Gordon equations arising in quantum field theory by using homotopy analysis transform method. Alex. Eng. J. 53, 469 (2014)
[51] Goswami, A, Singh, J, Kumar, D: A reliable algorithm for KdV equations arising in warm plasma. Nonlinear Eng. 5, 7 (2016)
[52] Xu, XX: A deformed reduced semi-discrete Kaup-Newell equation, the related integrable family and Darboux transformation. Appl. Math. Comput. 251, 275 (2015) · Zbl 1328.37054
[53] Zhao, QL, Li, XY, Liu, FS: Two integrable lattice hierarchies and their respective Darboux transformations. Appl. Math. Comput. 219, 5693 (2013) · Zbl 1288.37023
[54] Xu, XX, Sun, YP: An integrable coupling hierarchy of Dirac integrable hierarchy, its Liouville integrability and Darboux transformation. J. Nonlinear Sci. Appl. 10, 3328 (2017) · Zbl 1412.35300
[55] Dong, HH, Zhang, YF, Zhang, YF, Yin, BS: Generalized bilinear differential operators, binary Bell polynomials, and exact periodic wave solution of Boiti-Leon-Manna-Pempinelli equation. Abstr. Appl. Anal. 2014, 738609 (2014)
[56] Guo, XR: On bilinear representations and infinite conservation laws of a nonlinear variable-coefficient equation. Appl. Math. Comput. 248, 531 (2014) · Zbl 1338.37085
[57] Zhang, XE, Chen, Y, Zhang, Y: Breather, lump and X soliton solutions to nonlocal KP equation. Comput. Math. Appl. 74, 2341 (2017) · Zbl 1398.35210
[58] Nwozo, CR, Fadugba, SE: Some numerical methods for options valuation. Commun. Math. Finance 1, 51 (2012) · Zbl 1272.91126
[59] Hajipour, M, Malek, A: High accurate NRK and MWENO scheme for nonlinear degenerate parabolic PDEs. Appl. Math. Model. 36, 4439 (2012) · Zbl 1252.65166
[60] Hajipour, M, Malek, A: High accurate modified WENO method for the solution of Black-Scholes equation. Comput. Appl. Math. 34, 125-140 (2015) · Zbl 1314.91238
[61] Hajipour, M, Malek, A: Efficient high-order numerical methods for pricing of options. Comput. Econ. 45, 31 (2015)
[62] Elboree, MK: Derivation of soliton solutions to nonlinear evolution equations using He’s variational principle. Appl. Math. Model. 39, 4196 (2015) · Zbl 1443.35145
[63] Elboree, MK: Variational approach, soliton solutions and singular solitons for new coupled ZK system computers and mathematics with applications. Comput. Math. Appl. 70, 934 (2015) · Zbl 1443.35123
[64] Mohammed, KE: Conservation laws, soliton solutions for modified Camassa-Holm equation and \((2+1)(2+1)\)-dimensional ZK-BBM equation. Nonlinear Dyn. 89, 2979 (2017) · Zbl 1377.37095
[65] Zhang, JB, Ma, WX: Mixed lump-kink solutions to the BKP equation. Comput. Math. Appl. 74, 591 (2017) · Zbl 1387.35540
[66] Zhang, HQ, Ma, WX: Mixed lump-kink solutions to the KP equation. Comput. Math. Appl. 74, 1399 (2017) · Zbl 1394.35461
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.