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Time-fractional generalized Boussinesq equation for Rossby solitary waves with dissipation effect in stratified fluid and conservation laws as well as exact solutions. (English) Zbl 1426.76721

Summary: Construct fractional order model to describe Rossby solitary waves can provide more pronounced effects and deeper insight for comprehending generalization and evolution of Rossby solitary waves in stratified fluid. In the paper, from the quasi-geostrophic vorticity equation with dissipation effect and complete Coriolis force, based on the multi-scale analysis and perturbation method, a classical generalized Boussinesq equation is derived to describe the Rossby solitary waves in stratified fluid. Further, by employing the reduction perturbation method, the semi-inverse method, the Agrawal method, we derive the Euler-lagrangian equation of classical generalized Boussinesq equation and obtain the time-fractional generalized Boussinesq equation. Without dissipation effect, by using Lie group analysis method, the conservation laws of time-fractional Boussinesq equation are given. Finally, with the help of the improved \((G'/G)\) expansion method, the exact solutions of the above equation are generated. Meanwhile, in order to consider the dissipation effect, we have to derive the approximate solutions by adopting the New Iterative Method. We remark that the fractional order model can open up a new window for better understanding the waves in fluid.

MSC:

76U65 Rossby waves
35Q53 KdV equations (Korteweg-de Vries equations)
35C08 Soliton solutions
35Q35 PDEs in connection with fluid mechanics
35Q86 PDEs in connection with geophysics
35R11 Fractional partial differential equations
86A05 Hydrology, hydrography, oceanography
86A10 Meteorology and atmospheric physics
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