Modified Nonlinear Schrödinger Equation for Alfvén Waves Propagating along the Magnetic Field in Cold Plasmas

N-soliton solution for the derivative nonlinear Schrödinger equation with nonvanishing boundary conditions

“…Shifts of soliton positions due to collisions are analytically obtained, which are irrespective of the bright or dark characters of the participating solitons. The derivative nonlinear Schrödinger (DNLS) equation is an integrable model describing various nonlinear waves such as nonlinear Alfvén waves in space plasma(see, e.g., [1,2,3,4,5,6,7]), sub-picosecond pulses in single mode optical fibers(see, e.g., [8,9,10,11,12]), and weak nonlinear electromagnetic waves in ferromagnetic [13], antiferromagnetic [14], and dielectric[15] systems under external magnetic fields. Both of vanishing boundary conditions (VBC) and nonvanishing boundary conditions (NVBC) for the DNLS equation are physically significant.…”

DNLS equation for large-amplitude solitons propagating in an arbitrary direction in a high-β Hall plasma

“…This equation was first derived by Rogister (1971) starting with a Vlasov kinetic description for the particle species. Later it was derived by Mjølhus (1976) and Mio et al (1976) on the basis of Hall magnetohydrodynamics (MHD) for cold plasmas, and by Spangler and Sheerin (1982) and Sakai and Sonnerup (1983) from warm-fluid models. An excellent review of theory of quasiparallel small-amplitude nonlinear MHD waves based on the use of the DNLS equation and its generalizations has been given by Mjølhus and Hada (1997).…”

The hierarchy of higher order solutions of the derivative nonlinear Schrödinger equation

“…It is usually called the Peregrine soliton, and has been observed experimentally in fiber [25], water tank [26] and multi-component plasma [27]. The first order rogue wave solution of the DNLS equation was first found by Xu and coworkers [28] by the Darboux transformation 1 and certain limit technique. Recently, Guo et al [29] obtained two kinds of generalized Darboux transformations, and got the formulae of higher order solutions for both the VBC and NVBC.…”

“…It not only dominates the evolution of small-amplitude Alfén waves in a low-β plasma [1][2][3][4], but also is used to describe the behavior of large-amplitude magnetohydrodynamic (MHD) waves in a high-β plasma [5,6]. On the other hand, the DNLS equation governs the transmission of sub-picosecond in single mode optical fibers [7][8][9].…”