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Nonlinear Schrödinger equation for envelope Rossby waves with complete Coriolis force and its solution. (English) Zbl 1449.35399

Summary: The physical features of the equatorial envelope Rossby waves including with complete Coriolis force and dissipation are investigated analytically. Staring with a potential vorticity equation, the wave amplitude evolution of equatorial envelope Rossby waves is described as a nonlinear Schrödinger equation by employing multiple scale analysis and perturbation expansions. The equation is more suitable for describing envelope Rossby solitary waves when the horizontal component of Coriolis force is stronger near the equator. Then, based on the Jacobi elliptic function expansion method and trial function method, the classical Rossby solitary wave solution and the corresponding stream function of the envelope Rossby solitary waves are obtained, respectively. With the help of these solutions, the effect of dissipation and the horizontal component of Coriolis parameter are discussed in detail by graphical presentations. The results reveal the effect of the horizontal component of Coriolis force and dissipation on the classical Rossby solitary waves.

MSC:

35Q55 NLS equations (nonlinear Schrödinger equations)
35C20 Asymptotic expansions of solutions to PDEs
35C08 Soliton solutions
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[1] Barletti L, Brugnano L, Caccia GF et al (2018) Energy-conserving methods for the nonlinear Schrödinger equation. Appl Math Comput 318:3-18 · Zbl 1426.65202
[2] Belytschko T, Lu YY, Gu L (1994) Element-free Galerkin methods. Int J Numer Methods Eng 37:229-256 · Zbl 0796.73077
[3] Benney DJ (1979) Large amplitude Rossby waves. Stud Appl Math 6:01-10 · Zbl 0428.76025
[4] Benney DJ, Newell AC (1967) The propagation of nonlinear wave envelopes. J Math Phys 46:133-139 · Zbl 0153.30301
[5] Biazar J, Eslami M (2013) A new technique for non-linear two-dimensional wave equations. Sci Iran 20:359-363
[6] Boyd JP (1980) Equatorial solitary waves. Part I: Rossby solitons. J Phys Oceanogr 10:1699-1717
[7] Boyd JP (2018) Nonlinear equatorial waves. Berlin, Heidelberg
[8] Caillol P, Grimshaw RH (2008) Rossby elevation waves in the presence of a critical layer. Stud Appl Math 120:35-64 · Zbl 1386.76049
[9] Ching L, Sui CH, Yang MJ et al (2015) A modeling study on the effects of MJO and equatorial Rossby waves on tropical cyclone genesis over the western North Pacific in June 2004. Dyn Atmos Oceans 2:70-87
[10] Dellar PJ, Salmon R (2005) Shallow water equations with a complete Coriolis force and topography. Phys Fluids 17:106601 · Zbl 1188.76039
[11] Demiray H (2003) An analytical solution to the dissipative nonlinear Schrödinger equation. Appl Math Comput 145:179-184 · Zbl 1037.35077
[12] Eslami M (2014) An efficient method for solving fractional partial differential equations. Thai J Math 12:601-611 · Zbl 1328.35275
[13] Eslami M, Mirzazadeh M (2014) Study of convergence of Homotopy perturbation method for two-dimensional linear Volterra integral equations of the first kind. Int J Comput Sci Math 5:72-80 · Zbl 1312.65227
[14] Fu C, Lu CN, Yang HW (2018) Time – space fractional (2+1) dimensional nonlinear Schrödinger equation for envelope gravity waves in baroclinic atmosphere and conservation laws as well as exact solutions. Adv Differ Equ 1:56 · Zbl 1445.35281
[15] Guo M, Zhang Y, Wang M, Chen Y, Yang H (2018) A new ZK-ILW equation for algebraic gravity solitary waves in finite depth stratified atmosphere and the research of squall lines formation mechanism. Comput Math Appl. https://doi.org/10.1016/j.camwa.2018.02.019 · Zbl 1416.86008
[16] He JH (2003) Homotopy perturbation method: a new nonlinear analytical technique. Appl Math Comput 135:73-79 · Zbl 1030.34013
[17] Hua BL, Moore DW, Le Gentil S (1997) Inertial nonlinear equilibration of equatorial flows. J Fluid Mech 331:345-371 · Zbl 0910.76017
[18] Jeffrey A, Kawahara T (1982) Asymptotic methods in nonlinear wave theory. Applicable mathematics series. Pitman, Boston · Zbl 0473.35002
[19] Kasahara A (2003) On the nonhydrostatic atmospheric models with inclusion of the horizontal component of the Earth’s angular velocity. J Meteorol Soc Jpn 81:935-950
[20] Liu GT, Fan TY (2005) New applications of developed Jacobi elliptic function expansion methods. Phys Lett A 345:161-166 · Zbl 1345.35091
[21] Liu S, Fu Z, Liu S et al (2001) Jacobi elliptic function expansion method and periodic wave solutions of nonlinear wave equations. Phys Lett A 289:69-74 · Zbl 0972.35062
[22] Long RR (1964) Solitary waves in the Westerlies. J Atmos Sci 21:197-200
[23] Lu C, Fu C, Yang H (2018) Time-fractional generalized Boussinesq Equation for Rossby solitary waves with dissipation effect in stratified fluid and conservation laws as well as exact solutions. Appl Math Comput 327:104-116 · Zbl 1426.76721
[24] Luo DH (1991) Nonlinear Schrödinger equation in the rotational barotropic atmosphere and atmospheric blocking. Acta Meteorol Sin 5:587-590
[25] Philips NA (1968) Reply to G. Veronis’s comments on Phillips (1966). J Atmos Sci 25:1155-1157
[26] Puy M, Vialard J, Lengaigne M et al (2016) Modulation of equatorial Pacific westerly/easterly wind events by the Madden-Julian oscillation and convectively-coupled Rossby waves. Clim Dyn 46:2155-2178
[27] Quispel GRW, Nijhoff FW, Capel HW et al (1984) Bäklund transformations and singular integral equations. Phys A 123:319-359 · Zbl 0598.35103
[28] Raymond WH (2000) Equatorial meridional flows: rotationally induced circulations. Pure Appl Geophys 157:1767-1779
[29] Redekopp LG (1977) On the theory of solitary Rossby waves. J Fluid Mech 82:725-745 · Zbl 0362.76055
[30] Stewart AL, Dellar PJ (2010) Multilayer shallow water equations with complete Coriolis force. Part 1. Derivation on a non-traditional beta-plane. J Fluid Mech 651:387-413 · Zbl 1189.76098
[31] Stewart AL, Dellar PJ (2012) Multilayer shallow water equations with complete Coriolis force. Part 2. Linear plane waves. J Fluid Mech 690:16-50 · Zbl 1241.76093
[32] Tan B, Boyd JP (2000) Coupled-mode envelope solitary waves in a pair of cubic Schrödinger equations with cross modulation: analytical solution and collisions with application to Rossby waves. Chaos Solitons Fractals 11:1113-1129 · Zbl 0945.35086
[33] Triki H, Wazwaz AM (2011) Solitary wave solutions for a K (m, n, p, q + r) equation with generalized evolution. Int J Nonlinear Sci 11:387-395
[34] Triki H, Wazwaz AM (2014) Traveling wave solutions for fifth-order KdV type equations with time-dependent coefficients. Commun Nonlinear Sci Numer Simul 19:404-408 · Zbl 1440.35303
[35] Veronis G (1968) Comments on Phillips’s (1966) proposed simplification of the equations of motiofor shallow rotating atmosphere. J Atmos Sci 25:1154-1155
[36] Wadati M (1973) The modified Korteweg-deVries equation. J Phys Soc Jpn 34:1289-1296 · Zbl 1334.35299
[37] Wangsness RK (1970) Comments on “The equations of motion for a shallow rotating atmosphere and the ‘traditional approximation’”. J Atmos Sci 27:504-506
[38] White AA, Bromley RA (1995) Dynamically consistent, quasi-hydrostatic equations for global models with a complete representation of the Coriolis force. Q J R Meteorol Soc 121:399-418
[39] Yang HW, Wang XR, Yin BS (2014) A kind of new algebraic Rossby solitary waves generated by periodic external source. Nonlinear Dyn 76:1725-1735 · Zbl 1314.76019
[40] Yang HW, Chen X, Guo M et al (2018) A new ZK-BO equation for three-dimensional algebraic Rossby solitary waves and its solution as well as fission property. Nonlinear Dyn 91:2019-2032 · Zbl 1390.76044
[41] Yun-Long S, Hong-Wei Y, Bao-Shu Y et al (2015) Dissipative nonlinear Schrödinger equation with external forcing in rotational stratified fluids and its solution. Commun Theor Phys 64:464-472 · Zbl 1325.35206
[42] Zayed EME, Gepreel KA (2009) The (G′/G)-expansion method for finding traveling wave solutions of nonlinear partial differential equations in mathematical physics. J Math Phys 50:013502
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