Two-dimensional surface waves in magnetohydrodynamics

Abstract

The study of nonlinear waves in water has a long history beginning with the seminal paper by Korteweg & de Vries (Phil. Mag., vol. 39, 1895, p. 240) and more recently for magnetohydrodynamics Danov & Ruderman (Fluid Dyn., vol. 18, 1983, pp. 751–756). The appearance of a Hilbert transform in the nonlinear equation for magnetohydrodynamics (MHD) distinguishes it from the water wave model description. In this paper, we are interested in examining weakly nonlinear interfacial waves in $2+1$ dimensions. First, we determine the wave solution in the linear case. Next, we derive the corresponding generalisation for the Kadomtsev–Petviashvili (KP) equation with the inclusion of an equilibrium magnetic field. The derived governing equation is a generalisation of the Benjamin–Ono (BO) equation called the Benjamin equation first derived in Benjamin (J. Fluid Mech., vol. 245, 1992, pp. 401–411) and in the higher-dimensional context in Kim & Akylas (J. Fluid Mech., vol. 557, 2006, pp. 237–256).