Soliton Transmission Of The Derivative Nonlinear Schr(?)dinger Equation With Nonvanishing Boundary Conditions

The derivative nonlinear Schrodinger equation (DNLSE)is an integrable equation of many physical applications. It can model nonlinear Alfven waves in space plasma, sub-picosecond pulses in single mode optical fibers, nonlinear electromagnetic waves in dielectric and magnetic systems under external magnetic fields. The quantum DNLSE model is also applied in describing some one-dimensional chiral Luttinger liquids. Solutions of the DNLSE under both of the vanishing boundary conditions(VBC) and the nonvanishing boundary conditions (NVBC) are physically significant. For problems of nonlinear Alfven waves, weak nonlinear electromagnetic waves in magnetic and dielectric media, waves propagating strictly parallel to the ambient magnetic fields are modelled by the DNLSE with VBC while those oblique waves are modelled by the DNLSE with NVBC. For problems in optical fibers, pulses under bright background waves should be modelled by NVBC.For the DNLSE with VBC, inverse scattering transform(IST) and soliton solutions was achieved in the 70-80s last century. For the DNLSE with NVBC, a simple adapted IST was recently derived by introducing an affine parameter, yielding a much simpler one-soliton solution. All previous researches demonstrate that, in known (1+1) dimensional one-component integrable systems, the DNLSE with NVBC is a rare instance simultaneously admitting bright solitons, dark solitons, as well as their bound states(breathers). An explicit .N-soliton solution describing interactions between these solitons is thus in special demand. However, the first Lax equation for the DNLSE, unlike those in other integrable systems, is not an eigen equation of a linear operator, resulting in a potential-related phase η~+ which is determined by an integral about modulus of the soliton solution obtained by IST. For one-soliton solution, the integral was tackled by finding a special relation, which is obviously impossible to be extended to .N-soliton solution.In this dissertation, we derive an explicit pure .N-soliton solution for the DNLSE with NVBC, containing arbitrary number of bright and/or dark solitons. corresponding to purely imaginary discrete parameters. We get the modulus of the N-soliton solution and then the η~+ for pure two-soliton solution by using a special relation, which demonstrates a relation between η~+ and the denominator of the soliton solution. If this relation can be extended to arbitrary N, the shifts in soliton positions due to collisions are simply a summation of each two soliton collision. The three-soliton solution was numerically verified in the space-time domain when collision of three solitons simultaneously occurs. The .N-soliton solution thus appears to be valid. Details of collisions between two solitons and among three solitons are graphically shown. Interesting characteristics between bright and dark solitons are found. They are consistent with those in breathers, demonstrating that the breather is actually a bounded pair of bright and dark solitons. Finally, we find the first conservation law should be modified by adding a phase.