Open Access. Powered by Scholars. Published by Universities.^{®}

Articles **1** - **4** of ** 4**

## Full-Text Articles in Physics

Asymptotic Solitons For A Higher-Order Modified Korteweg–De Vries Equation, T. Marchant

#### Asymptotic Solitons For A Higher-Order Modified Korteweg–De Vries Equation, T. Marchant

*Tim Marchant*

Solitary wave interaction for a higher-order modified Korteweg–de Vries (mKdV) equation is examined. The higher-order mKdV equation can be asymptotically transformed to the mKdV equation, if the higher-order coefficients satisfy a certain algebraic relationship. The transformation is used to derive the higher-order two-soliton solution and it is shown that the interaction is asymptotically elastic. Moreover, the higher-order phase shifts are derived using the asymptotic theory. Numerical simulations of the interaction of two higher-order solitary waves are also performed. Two examples are considered, one satisfies the algebraic relationship derived from the asymptotic theory, and the other does not. For the ...

High-Order Interaction Of Solitary Waves On Shallow Water, Prof. Tim Marchant

#### High-Order Interaction Of Solitary Waves On Shallow Water, Prof. Tim Marchant

*Tim Marchant*

The interaction of solitary waves on shallow water is examined to fourth order. At first order the interaction is governed by the Korteweg-de Vries (KdV) equation, and it is shown that the unidirectional assumption, of right-moving waves only, is incompatible with mass conservation at third order. To resolve this, a mass conserving system of KdV equations, involving both right- and left-moving waves, is derived to third order. A fourth-order interaction term, in which the right- and left-moving waves are coupled, is also derived as this term is crucial in determining the fourth-order change in solitary wave amplitude. The form of ...

The Initial Boundary Problem For The Korteweg-De Vries Equation On The Negative Quarter-Plane, Prof. Tim Marchant

#### The Initial Boundary Problem For The Korteweg-De Vries Equation On The Negative Quarter-Plane, Prof. Tim Marchant

*Tim Marchant*

The initial boundary-value problem for the Korteweg-de Vries (KdV) equation on the negative quarter-plane, x < 0 and t > 0, is considered. The formulation of this problem is different to the usual initial boundary-value problem on the positive quarter-plane, for which x > 0 and t > 0. Two boundary conditions are required at x = 0 for the negative quarter-plane problem, in contrast to the one boundary condition needed at x = 0 for the positive quarter-plane problem. Solutions of the KdV equation on the infinite line, such as the soliton, cnoidal wave, mean height variation and undular bore solution, are used to find approximate solutions to ...

Numerical Solitary Wave Interaction: The Order Of The Inelastic Effect, Prof. Tim Marchant

#### Numerical Solitary Wave Interaction: The Order Of The Inelastic Effect, Prof. Tim Marchant

*Tim Marchant*

Solitary wave interaction is examined using an extended Benjamin-Bona-Mahony (eBBM) equation. This equation includes higher-order nonlinear and dispersive effects and is is asymptotically equivalent to the extended Korteweg-de Vries (eKdV) equation. The eBBM formulation is preferable to the eKdV equation for the numerical modelling of solitary wave collisions, due to the stability of its finite-difference scheme. In particular, it allows the interaction of steep waves to be modelled, which due to numerical instability, is not possible using the eKdV equation. Numerical simulations of a number of solitary wave collisions of varying nonlinearity are performed for two special cases corresponding to ...