# American Institute of Mathematical Sciences

August  2017, 37(8): 4391-4398. doi: 10.3934/dcds.2017188

## Exact azimuthal internal waves with an underlying current

 Department of Marine Environment and Engineering, National Sun Yat-sen University, Kaohsiung 80424, Taiwan

Received  January 2017 Revised  May 2017 Published  April 2017

In this paper, we present an explicit and exact solution of the nonlinear governing equations including Coriolis and centripetal terms for internal azimuthal waves with a uniform current in the $\beta$-plane approximation near the equator. This solution is described in the Lagrangian framework. The unidirectional azimuthal internal trapped are symmetric about the equator and propagate eastward above the thermocline and beneath the near-surface layer.

Citation: Hung-Chu Hsu. Exact azimuthal internal waves with an underlying current. Discrete & Continuous Dynamical Systems - A, 2017, 37 (8) : 4391-4398. doi: 10.3934/dcds.2017188
##### References:

show all references

##### References:
 [1] Jifeng Chu, Delia Ionescu-Kruse, Yanjuan Yang. Exact solution and instability for geophysical waves at arbitrary latitude. Discrete & Continuous Dynamical Systems - A, 2019, 39 (8) : 4399-4414. doi: 10.3934/dcds.2019178 [2] Tony Lyons. Geophysical internal equatorial waves of extreme form. Discrete & Continuous Dynamical Systems - A, 2019, 39 (8) : 4471-4486. doi: 10.3934/dcds.2019183 [3] Mateusz Kluczek. Nonhydrostatic Pollard-like internal geophysical waves. Discrete & Continuous Dynamical Systems - A, 2019, 39 (9) : 5171-5183. doi: 10.3934/dcds.2019210 [4] Út V. Lê. Regularity of the solution of a nonlinear wave equation. Communications on Pure & Applied Analysis, 2010, 9 (4) : 1099-1115. doi: 10.3934/cpaa.2010.9.1099 [5] Umberto De Maio, Akamabadath K. Nandakumaran, Carmen Perugia. Exact internal controllability for the wave equation in a domain with oscillating boundary with Neumann boundary condition. Evolution Equations & Control Theory, 2015, 4 (3) : 325-346. doi: 10.3934/eect.2015.4.325 [6] Jibin Li, Yi Zhang. Exact solitary wave and quasi-periodic wave solutions for four fifth-order nonlinear wave equations. Discrete & Continuous Dynamical Systems - B, 2010, 13 (3) : 623-631. doi: 10.3934/dcdsb.2010.13.623 [7] Ferhat Mohamed, Hakem Ali. Energy decay of solutions for the wave equation with a time-varying delay term in the weakly nonlinear internal feedbacks. Discrete & Continuous Dynamical Systems - B, 2017, 22 (2) : 491-506. doi: 10.3934/dcdsb.2017024 [8] Jibin Li, Fengjuan Chen. Exact travelling wave solutions and their dynamical behavior for a class coupled nonlinear wave equations. Discrete & Continuous Dynamical Systems - B, 2013, 18 (1) : 163-172. doi: 10.3934/dcdsb.2013.18.163 [9] José Raúl Quintero, Juan Carlos Muñoz Grajales. Solitary waves for an internal wave model. Discrete & Continuous Dynamical Systems - A, 2016, 36 (10) : 5721-5741. doi: 10.3934/dcds.2016051 [10] Zhaosheng Feng, Qingguo Meng. Exact solution for a two-dimensional KDV-Burgers-type equation with nonlinear terms of any order. Discrete & Continuous Dynamical Systems - B, 2007, 7 (2) : 285-291. doi: 10.3934/dcdsb.2007.7.285 [11] Biswajit Basu. On an exact solution of a nonlinear three-dimensional model in ocean flows with equatorial undercurrent and linear variation in density. Discrete & Continuous Dynamical Systems - A, 2019, 39 (8) : 4783-4796. doi: 10.3934/dcds.2019195 [12] Claudianor O. Alves. Existence of periodic solution for a class of systems involving nonlinear wave equations. Communications on Pure & Applied Analysis, 2005, 4 (3) : 487-498. doi: 10.3934/cpaa.2005.4.487 [13] Kateryna Marynets. Study of a nonlinear boundary-value problem of geophysical relevance. Discrete & Continuous Dynamical Systems - A, 2019, 39 (8) : 4771-4781. doi: 10.3934/dcds.2019194 [14] Kaili Zhuang, Tatsien Li, Bopeng Rao. Exact controllability for first order quasilinear hyperbolic systems with internal controls. Discrete & Continuous Dynamical Systems - A, 2016, 36 (2) : 1105-1124. doi: 10.3934/dcds.2016.36.1105 [15] Jibin Li, Yan Zhou. Bifurcations and exact traveling wave solutions for the nonlinear Schrödinger equation with fourth-order dispersion and dual power law nonlinearity. Discrete & Continuous Dynamical Systems - S, 2018, 0 (0) : 0-0. doi: 10.3934/dcdss.2020113 [16] Jiao Chen, Weike Wang. The point-wise estimates for the solution of damped wave equation with nonlinear convection in multi-dimensional space. Communications on Pure & Applied Analysis, 2014, 13 (1) : 307-330. doi: 10.3934/cpaa.2014.13.307 [17] R. Bartolo, Anna Maria Candela, J.L. Flores, Addolorata Salvatore. Periodic trajectories in plane wave type spacetimes. Conference Publications, 2005, 2005 (Special) : 77-83. doi: 10.3934/proc.2005.2005.77 [18] Alan Compelli, Rossen Ivanov. Benjamin-Ono model of an internal wave under a flat surface. Discrete & Continuous Dynamical Systems - A, 2019, 39 (8) : 4519-4532. doi: 10.3934/dcds.2019185 [19] Viorel Barbu, Ionuţ Munteanu. Internal stabilization of Navier-Stokes equation with exact controllability on spaces with finite codimension. Evolution Equations & Control Theory, 2012, 1 (1) : 1-16. doi: 10.3934/eect.2012.1.1 [20] Barbara Abraham-Shrauner. Exact solutions of nonlinear partial differential equations. Discrete & Continuous Dynamical Systems - S, 2018, 11 (4) : 577-582. doi: 10.3934/dcdss.2018032

2018 Impact Factor: 1.143