Mathematics and Statistics 9(5): 825-834, 2021 DOI: 10.13189/ms.2021.090523 http://www.hrpub.org Numerical Solution of Ostrovsky Equation over Variable Topography Passes through Critical Point Using Pseudospectral Method Nik Nur Amiza Nik Ismail, Azwani Alias ∗ , Fatimah Noor Harun Faculty of Ocean Engineering Technology and Informatics, Universiti Malaysia Terengganu, 21030 Kuala Nerus, Terengganu, Malaysias Received July 5, 2021; Revised August 20, 2021; Accepted September 21, 2021 Cite This Paper in the following Citation Styles (a): [1] Nik Nur Amiza Nik Ismail, Azwani Alias, Fatimah Noor Harun, ”Numerical Solution of Ostrovsky Equation over Variable Topography Passes through Critical Point Using Pseudospectral Method,” Mathematics and Statistics, Vol.9, No.5, pp. 825-834, 2021. DOI: 10.13189/ms.2021.090523 (b): Nik Nur Amiza Nik Ismail, Azwani Alias, Fatimah Noor Harun, (2021). Numerical Solution of Ostrovsky Equation over Variable Topography Passes through Critical Point Using Pseudospectral Method. Mathematics and Statistics, 9(5), 825-834. DOI: 10.13189/ms.2021.090523 Copyright ©2021 by authors, all rights reserved. Authors agree that this article remains permanently open access under the terms of the Creative Commons Attribution License 4.0 International License Abstract Internal solitary waves have been documented in several parts of the world. This paper intends to look at the effects of the variable topography and rotation on the evolution of the internal waves of depression. Here, the wave is considered to be propagating in a two-layer fluid system with the background topography is assumed to be rapidly and slowly varying. Therefore, the appropriate mathematical model to describe this situation is the variable-coefficient Ostrovsky equation. In particular, the study is interested in the transition of the internal solitary wave of depression when there is a polarity change under the influence of background rotation. The numerical results using the Pseudospectral method show that, over time, the internal solitary wave of elevation transforms into the internal solitary wave of depression as it propagates down a decreasing slope and changes its polarity. However, if the background rotation is considered, the internal solitary waves decompose and form a wave packet and its envelope amplitude decreases slowly due to the decreasing bottom surface. The numerical solutions show that the combination effect of variable topography and rotation when passing through the critical point affected the features and speed of the travelling solitary waves. Keywords Solitary Wave, Nonlinear Equation, Ostrovsky Equation, Variable Topography, Background Rotation, Polarity Change, Pseudospectral Method 1 Introduction The classical model for weakly nonlinear waves propagat- ing over a uniform topography, h, yields to the formation of the Korteweg–de Vries (KdV) equation. The effect of the variable topography, however, must be taken into considera- tion when retrieving the mathematical model. In consequence, the KdV is succeeded by the variable-coefficient Korteweg–de Vries (vKdV) equation. Johnson [1] was the first person who derive the vKdV equation for surface waves, in which Q = c, and were discussed recently by Grimshaw et al. [2] for inter- nal waves. In the form of the vKdV equation, Grimshaw [3, 4] carried out a detailed analysis and an effective model of soli- tary wave propagation over the variable topography. On the assumption that the flow is two-dimensional, with x and z in- dicating horizontal and vertical coordinates, respectively, the end result is A t + cA x + cQ x 2Q A + μAA x + λA xxx =0, (1) where A(x, t) is the amplitude of the modal function φ(z), de- fined by {ρ 0 (c − u 0 ) 2 φ z } z + ρ 0 N 2 φ =0, for − h<z< 0, φ =0, at z = −h, (c − u 0 ) 2 φ z = gφ, at z =0, that can also be utilized to find the linear phase speed, c. Here, g is gravitational acceleration, ρ 0 (z) is the density of the back- ground that can be defined by the buoyancy frequency N (z) in which, ρ 0 N 2 = −gρ 0z and u 0 (z) represent the background current, while t is time variables. The coefficient μ in equation (1) represents the nonlinear term, while λ is the coefficient of