Coupled KdV equations derived from atmospherical dynamics

Some types of coupled Korteweg de-Vries (KdV) equations are derived from an atmospheric dynamical system. In the derivation procedure, an unreasonable $y$-average trick (which is usually adopted in literature) is removed. The derived models are classified via Painlev\'e test. Three types of $\tau$-function solutions and multiple soliton solutions of the models are explicitly given by means of the exact solutions of the usual KdV equation. It is also interesting that for a non-Painlev\'e integrable coupled KdV system there may be multiple soliton solutions.Comment: 19 pages, 2 figure

Mathematical modelling of nonlinear internal waves in a rotating fluid

Large amplitude internal solitary waves in the coastal ocean are commonly modelled with the Korteweg-de Vries (KdV) equation or a closely related evolution equation. The characteristic feature of these models is the solitary wave solution, and it is well documented that these provide the basic paradigm for the interpretation of oceanic observations. However, often internal waves in the ocean survive for several inertial periods, and in that case, the KdV equation is supplemented with a linear non-local term representing the effects of background rotation, commonly called the Ostrovsky equation. This equation does not support solitary wave solutions, and instead a solitary-like initial condition collapses due to radiation of inertia-gravity waves, with instead the long-time outcome typically being an unsteady nonlinear wave packet. The KdV equation and the Ostrovsky equation are formulated on the assumption that only a single vertical mode is used. In this thesis we consider the situation when two vertical modes are used, due to a near-resonance between their respective linear long wave phase speeds. This phenomenon can be described by a pair of coupled Ostrovsky equations, which is derived asymptotically from the full set of Euler equations and solved numerically using a pseudo-spectral method. The derivation of a system of coupled Ostrovsky equations is an important extension of coupled KdV equations on the one hand, and a single Ostrovsky equation on the other hand. The analytic structure and dynamical behaviour of the system have been elucidated in two main cases. The first case is when there is no background shear flow, while the second case is when the background state contains current shear, and both cases lead to new solution types with rich dynamical behaviour. We demonstrate that solitary-like initial conditions typically collapse into two unsteady nonlinear wave packets, propagating with distinct speeds corresponding to the extremum value in the group velocities. However, a background shear flow allows for several types of dynamical behaviour, supporting both unsteady and steady nonlinear wave packets, propagating with the speeds which can be predicted from the linear dispersion relation. In addition, in some cases secondary wave packets are formed associated with certain resonances which also can be identified from the linear dispersion relation. Finally, as a by-product of this study it was shown that a background shear flow can lead to the anomalous version of the single Ostrovsky equation, which supports a steady wave packet

A new approach for numerical solution of modified korteweg-de vries equation

In this paper, a lumped Galerkin method is applied with cubic B-spline interpolation functions to find the numerical solution of the modified Korteweg-de Vries (mKdV) equation. Test problems including motion of single solitary wave, interaction of two solitons, interaction of three solitons, and evolution of solitons are solved to verify the proposed method by calculating the error norms L2 and L1 and the conserved quantities mass, momentum and energy. Applying the von-Neumann stability analysis, the proposed method is shown to be unconditionally stable. Consequently, the obtained results are found to be harmony with the some recent result