On Numerical Solution of the Gardner–Ostrovsky Equation

Abstract

Abstract. A simple explicit numerical scheme is proposed for the solution of the GardnerOstrovsky equation ut + cux + αuux + α1u 2 ux + βuxxx x = γu which is also known as the extended rotation-modified Korteweg-de Vries (KdV) equation. This equation is used for the description of internal oceanic waves affected by Earth' rotation. Particular versions of this equation with zero some of coefficients, α, α1, β, or γ are also known in numerous applications. The proposed numerical scheme is a further development of the well-known finite-difference scheme earlier used for the solution of the KdV equation. The scheme is of the second order accuracy both on temporal and spatial variables. The stability analysis of the scheme is presented for infinitesimal perturbations. The conditions for the calculations with the appropriate accuracy have been found. Examples of calculations with the periodic boundary conditions are presented to illustrate the robustness of the proposed scheme.

Validity of the Weakly Nonlinear Solution of the Cauchy Problem for the Boussinesq‐Type Equation

“…Bound (35) is proved similarly after writingÛ = U 2 + Ǔ , where U 2 is a bounded solution of system (30), whereas φ ± in (27) satisfy the linearized KdV equations (32) subject to the initial data (29). The new error termǓ satisfies a priori energy estimate similar to (42) with the new error term.…”

Formation of wave packets in the Ostrovsky equation for both normal and anomalous dispersion

“…method (for details, see [6,23] and the references therein) to find the steady wave solutions. The obtained numerical results are shown in figure 2 by dots.…”

“…These numerical simulations were performed using the finite-difference numerical scheme described in [23] with periodic boundary conditions on the interval of length L. In the first run which can be treated as the reference case studied in [23], we put the following coefficients in equation ( For this case γ > 0, we recall the well-established results from [3,13,23]. The solitary wave in the course of propagation experiences terminal decay at the early stage of evolution in accordance with equation (3.12).…”

Solitary Waves, Shock Waves and Singular Solitons of the Generalized Ostrovsky-Benjamin-Bona-Mahoney Equation

“…The Ostrovsky equation arises in the study of geophysical fluids while the Benjamin-Bona-Mahoney (BBM) equation is studied in the context of shallow water wave dynamics [3,14,15,16,17]. This equation models the dynamics that is much closer to realistic situations.…”

“…The theory of solitons plays a dominant role in many nonlinear dynamical systems [1,2,3,4,5]. Of particular interest is in the field of nonlinear optical comminications systems.…”