Abstract
We present the full classification of wave patterns evolving from an initial step-like discontinuity for arbitrary choice of boundary conditions at the discontinuity location in the DNLS equation theory. In this non-convex dispersive hydrodynamics problem, solutions of the Whitham modulation equations are mapped to parameters of a modulated wave by two-valued functions what makes situation much richer than that for a convex case of the NLS equation type. In particular, new types of simple-wave-like structures, such as trigonometric shocks and combined shocks, appear as building elements of the whole wave pattern observable in the long-time evolution of pulses after the wave-breaking point. The developed theory can find applications to propagation of light pulses in fibers and to the theory of Alfvén dispersive shock waves.
Highlights
The problem of classification of wave structures evolving from initial discontinuities has played important role since the classical paper of B
This approach seems very natural from physical point of view since it provides some information on the inner structure of viscous shocks. There exists another method of regularization of hydrodynamics-like equations, namely, introduction of small dispersion. In this case the limit of zero dispersion does not lead to the same shock structure, this approach is of considerable interest since, on one side, it is related with the theory of dispersive shock waves (DSWs) that finds a number of physical applications and, on another hand, there are situations when the regularized equation belongs to the class of completely integrable equations and it admits quite thorough investigation including even cases of non-genuinely nonlinear hyperbolic systems
We have developed the Whitham method of modulations for evolution of waves governed by the derivative nonlinear Schrodinger (DNLS) equation
Summary
The problem of classification of wave structures evolving from initial discontinuities has played important role since the classical paper of B. For classification of wave patterns arising in solutions of the Riemann problem of the KdV or NLS type, it is important that the corresponding dispersionless limits (Hopf equation or shallow water equations) are represented by the genuinely nonlinear hyperbolic equations. If it is not the case, the classification of the KdV-NLS type becomes insufficient and it was found that it should include new elements— kinks or trigonometric dispersive shocks—for mKdV [18] and Miyata-Camassa-Choi [19] equations. We develop a similar theory for the equation (1)
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
