We address existence of global solutions to the derivative nonlinear Schrödinger (DNLS) equation without the small-norm assumption. By using the inverse scattering transform method without eigenvalues and resonances, we construct a unique global solution in H 2 (R) ∩ H 1,1 (R) which is also Lipschitz continuous with respect to the initial data. Compared to the existing literature on the spectral problem for the DNLS equation, the corresponding Riemann-Hilbert problem is defined in the complex plane with the jump on the real line. 4 ,We also note another useful elementary result.Using Cauchy-Schwarz inequality for r − (z) ∈ H 1 z (R) ∩ L 2,1 z (R), we obtain the desired bound.

We prove H 1 orbital stability of Dirac solitons in the integrable massive Thirring model by working with an additional conserved quantity which complements Hamiltonian, momentum and charge functionals of the general nonlinear Dirac equations. We also derive a global bound on the H 1 norm of the L 2 -small solutions of the massive Thirring model.

Abstract. We prove the global existence result for the derivative NLS equation in the case when the initial datum includes a finite number of solitons. This is achieved by an application of the Bäcklund transformation that removes a finite number of zeros of the scattering coefficient. By means of this transformation, the Riemann-Hilbert problem for meromorphic functions can be formulated as the one for analytic functions, the solvability of which was obtained recently. A major difficulty in the proof is to show invertibility of the Bäcklund transformation acting on weighted Sobolev spaces.

We prove L 2 orbital stability of Dirac solitons in the massive Thirring model. Our analysis uses local well posedness of the massive Thirring model in L 2 , conservation of the charge functional, and the auto-Bäcklund transformation. The latter transformation exists because the massive Thirring model is integrable via the inverse scattering transform method.

The advent of nanotechnology has facilitated the preparation of colloidal particles with adjustable sizes and the control of their size-dependent properties. Physical manipulation, such as separation, mixing, and concentration, of these colloidal particles represents an essential step for fully utilizing their potential in a wide spectrum of nanotechnology applications. In this study, we investigate hybrid electrokinetics, the combination of dielectrophoresis and electrohydrodynamics, for active manipulation of colloidal particles ranging from nanometers to micrometers in size. A concentric electrode configuration, which is optimized for generating electrohydrodynamic flow, has been designed to elucidate the effectiveness of hybrid electrokinetics and define the operating regimes for different microfluidic operations. The results indicate that the relative importance of electrohydrodynamics increases with decreasing particle size as predicted by a scaling analysis and that electrohydrodynamics is pivotal for manipulating nanoscale particles. Using the concentric electrodes, we demonstrate separation, mixing, and concentration of colloidal particles by adjusting the relative strengths of different electrokinetic phenomena. The effectiveness of hybrid electrokinetics indicates its potential to serve as a generic technique for active manipulation of colloidal particles in various nanotechnology applications.