Integrable discretizations of derivative nonlinear Schroedinger equations

We propose integrable discretizations of derivative nonlinear Schroedinger (DNLS) equations such as the Kaup-Newell equation, the Chen-Lee-Liu equation and the Gerdjikov-Ivanov equation by constructing Lax pairs. The discrete DNLS systems admit the reduction of complex conjugation between two dependent variables and possess bi-Hamiltonian structure. Through transformations of variables and reductions, we obtain novel integrable discretizations of the nonlinear Schroedinger (NLS), modified KdV (mKdV), mixed NLS, matrix NLS, matrix KdV, matrix mKdV, coupled NLS, coupled Hirota, coupled Sasa-Satsuma and Burgers equations. We also discuss integrable discretizations of the sine-Gordon equation, the massive Thirring model and their generalizations.Comment: 24 pages, LaTeX2e (IOP style), final versio

A new numerical application of the generalized Rosenau-RLW equation

. This study implemented a collocation nite element method based on septic B-splines as a tool to obtain the numerical solutions of the nonlinear generalized RosenauRLW equation. One of the advantages of this method is that when the bases are chosen at a high degree, better numerical solutions are obtained. E ectiveness of the method is demonstrated by solving the equation with various initial and boundary conditions. Further, in order to detect the performance of the method, L2 and L1 error norms and two lowest invariants IM and IE were computed. The obtained numerical results were compared with some of those in the literature for similar parameters. This comparison clearly shows that the obtained results are better than and in good conformity with some of the earlier results. Stability analysis demonstrates that the proposed algorithm, based on a Crank Nicolson approximation in time, is unconditionally stable

Long's Vortex Revisited

The conical self-similar vortex solution of Long (1961) is reconsidered, with a view toward understanding what, if any, relationship exists between Long's solution and the more-recent similarity solutions of Mayer and Powell (1992), which are a rotational-flow analogue of the Falkner-Skan boundary-layer flows, describing a self-similar axisymmetric vortex embedded in an external stream whose axial velocity varies as a power law in the axial (z) coordinate, with phi=r/z^n being the radial similarity coordinate and n the core growth rate parameter. We show that, when certain ostensible differences in the formulations and radial scalings are properly accounted for, the Long and Mayer-Powell flows in fact satisfy the same system of coupled ordinary differential equations, subject to different kinds of outer-boundary conditions, and with Long's equations a special case corresponding to conical vortex core growth, n=1 with outer axial velocity field decelerating in a 1/z fashion, which implies a severe adverse pressure gradient. For pressure gradients this adverse Mayer and Powell were unable to find any leading-edge-type vortex flow solutions which satisfy a basic physicality criterion based on monotonicity of the total-pressure profile of the flow, and it is shown that Long's solutions also violate this criterion, in an extreme fashion. Despite their apparent nonphysicality, the fact that Long's solutions fit into a more general similarity framework means that nonconical analogues of these flows should exist. The far-field asymptotics of these generalized solutions are derived and used as the basis for a hybrid spectral-numerical solution of the generalized similarity equations, which reveal the existence of solutions for more modestly adverse pressure gradients than those in Long's case, and which do satisfy the above physicality criterion.Comment: 30 pages, including 16 figure

Stochastic continuity equations with conservative noise

The present article is devoted to well-posedness by noise for the continuity equation. Namely, we consider the continuity equation with non-linear and partially degenerate stochastic perturbations in divergence form. We prove the existence and uniqueness of entropy solutions under hypotheses on the velocity field which are weaker than those required in the deterministic setting. This extends related results of [Flandoli, Gubinelli, Priola; Invent. Math., 2010] applicable for linear multiplicative noise to a non-linear setting. The existence proof relies on a duality argument which makes use of the regularity theory for fully non-linear parabolic equations.Comment: 42 page

Rogue wave solutions for an inhomogeneous fifth-order nonlinear Schrodinger equation from Heisenberg ferromagnetism

In this paper, generalized Darboux transformation for an inhomogeneous fifth-order nonlinear Schrodinger equation from Heisenberg ferromagnetism are constructed according to which rouge wave solutions of the equation are obtained. Influences of equation parameter on the evolution of rogue waves are discussed. With the aid of Mathematica, some special solutions are graphically illustrated which could help to better understand the evolution of rogue waves