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An Introduction to Quasiparallel Alfven Waves

My primary research interest is in the nonlinear dynamics of Alfven waves. These waves play an important role in the transport and acceleration of protons in the solar wind that, in turn, can adversely effect satellites, radio communications and even power grids. The focus of my work lies in the theoretical investigation of these waves. The character of these waves is determined in part by their travel along the sun’s magnetic field. The waves become particularly intriguing when they travel parallel, or at a small angle, to the magnetic field. Here they display a nonlinear coupling between two otherwise distinct wave modes. In this regime the waves are described by the Derivative Nonlinear Schrodinger (DNLS) equation. For parallel propagation (and large wave numbers) they are described by the Nonlinear Schrodinger (NLS) equation and at larger angles the modes become distinct, as they travel at very different speeds at such angles, and one mode is described by the Korteweg-deVries (KdV) while the other is described by the Modified Korteweg-deVries (MKdV) equation. Normally, one would not expect to be able to find general solutions to nonlinear partial differential equations. It is remarkable that each of these equations can be solved exactly with a procedure known as the Inverse Scattering Transformation (IST). What makes the DNLS equation intriguing is that it contains each of the other three equations as special limiting cases. The DNLS contains within it the nonlinear dynamics of the other equations as well as dynamical effects that are transitional between them. The dynamics of Alfven waves, and their description by the DNLS equation, are particularly rich and complex.