New variable separation approach: application to nonlinear diffusion equations

The concept of the derivative-dependent functional separable solution, as a generalization to the functional separable solution, is proposed. As an application, it is used to discuss the generalized nonlinear diffusion equations based on the generalized conditional symmetry approach. As a consequence, a complete list of canonical forms for such equations which admit the derivative-dependent functional separable solutions is obtained and some exact solutions to the resulting equations are described.Comment: 19 pages, 2 fig

Approximate perturbed direct homotopy reduction method: infinite series reductions to two perturbed mKdV equations

An approximate perturbed direct homotopy reduction method is proposed and applied to two perturbed modified Korteweg-de Vries (mKdV) equations with fourth order dispersion and second order dissipation. The similarity reduction equations are derived to arbitrary orders. The method is valid not only for single soliton solution but also for the Painlev\'e II waves and periodic waves expressed by Jacobi elliptic functions for both fourth order dispersion and second order dissipation. The method is valid also for strong perturbations.Comment: 8 pages, 1 figur

A multiple exp-function method for nonlinear differential equations and its application

A multiple exp-function method to exact multiple wave solutions of nonlinear partial differential equations is proposed. The method is oriented towards ease of use and capability of computer algebra systems, and provides a direct and systematical solution procedure which generalizes Hirota's perturbation scheme. With help of Maple, an application of the approach to the $3+1$ dimensional potential-Yu-Toda-Sasa-Fukuyama equation yields exact explicit 1-wave and 2-wave and 3-wave solutions, which include 1-soliton, 2-soliton and 3-soliton type solutions. Two cases with specific values of the involved parameters are plotted for each of 2-wave and 3-wave solutions.Comment: 12 pages, 16 figure

A Model for the Moving `Wisps' in the Crab Nebula

I propose that the moving `wisps' near the center of the Crab Nebula result from nonlinear Kelvin-Helmholtz instabilities in the equatorial plane of the shocked pulsar wind. Recent observations suggest that the wisps trace out circular wavefronts in this plane, expanding radially at speeds approximately less than c/3. Instabilities could develop if there is sufficient velocity shear between a faster-moving equatorial zone and a slower moving shocked pulsar wind at higher latitudes. The development of shear could be related to the existence of a neutral sheet -- with weak magnetic field -- in the equatorial zone, and could also be related to a recent suggestion by Begelman that the magnetic field in the Crab pulsar wind is much stronger than had been thought. I show that plausible conditions could lead to the growth of instabilities at the radii and speeds observed, and that their nonlinear development could lead to the appearance of sharp wisplike features.Comment: 7 pages; 3 postscript figures; LaTex, uses emulateapj.sty; to Appear in the Astrophysical Journal, Feb. 20, 1999, Vol. 51

From nothing to something: discrete integrable systems

Chinese ancient sage Laozi said that everything comes from `nothing'. Einstein believes the principle of nature is simple. Quantum physics proves that the world is discrete. And computer science takes continuous systems as discrete ones. This report is devoted to deriving a number of discrete models, including well-known integrable systems such as the KdV, KP, Toda, BKP, CKP, and special Viallet equations, from `nothing' via simple principles. It is conjectured that the discrete models generated from nothing may be integrable because they are identities of simple algebra, model-independent nonlinear superpositions of a trivial integrable system (Riccati equation), index homogeneous decompositions of the simplest geometric theorem (the angle bisector theorem), as well as the M\"obious transformation invariants.Comment: 11 pages, side 10 repor

Dynamic Evolution of a Quasi-Spherical General Polytropic Magnetofluid with Self-Gravity

In various astrophysical contexts, we analyze self-similar behaviours of magnetohydrodynamic (MHD) evolution of a quasi-spherical polytropic magnetized gas under self-gravity with the specific entropy conserved along streamlines. In particular, this MHD model analysis frees the scaling parameter $n$ in the conventional polytropic self-similar transformation from the constraint of $n+\gamma=2$ with $\gamma$ being the polytropic index and therefore substantially generalizes earlier analysis results on polytropic gas dynamics that has a constant specific entropy everywhere in space at all time. On the basis of the self-similar nonlinear MHD ordinary differential equations, we examine behaviours of the magnetosonic critical curves, the MHD shock conditions, and various asymptotic solutions. We then construct global semi-complete self-similar MHD solutions using a combination of analytical and numerical means and indicate plausible astrophysical applications of these magnetized flow solutions with or without MHD shocks.Comment: 21 pages, 7 figures, accepted for publication in APS

MHD tidal waves on a spinning magnetic compact star

In an X-ray binary system, the companion star feeds the compact neutron star with plasma materials via accretions. The spinning neutron star is likely covered with a thin "magnetized ocean" and may support {\it magnetohydrodynamic (MHD) tidal waves}. While modulating the thermal properties of the ocean, MHD tidal waves periodically shake the base of the stellar magnetosphere that traps energetic particles, including radiating relativistic electrons. For a radio pulsar, MHD tidal waves in the stellar surface layer may modulate radio emission processes and leave indelible signatures on timescales different from the spin period. Accretion activities are capable of exciting these waves but may also obstruct or obscure their detections meanwhile. Under fortuitous conditions, MHD tidal waves might be detectable and offer valuable means to probe properties of the underlying neutron star. Similar situations may also occur for a cataclysmic variable -- an accretion binary system that contains a rotating magnetic white dwarf. This Letter presents the theory for MHD tidal waves in the magnetized ocean of a rotating degenerate star and emphasizes their potential diagnostics in X-ray and radio emissions.Comment: ApJ Letter paper already publishe

Ultrafast Hole Trapping and Relaxation Dynamics in p-Type CuS Nanodisks

CuS nanocrystals are potential materials for developing low-cost solar energy conversion devices. Understanding the underlying dynamics of photoinduced carriers in CuS nanocrystals is essential to improve their performance in these devices. In this work, we investigated the photoinduced hole dynamics in CuS nanodisks (NDs) using the combination of transient optical (OTA) and X-ray (XTA) absorption spectroscopy. OTA results show that the broad transient absorption in the visible region is attributed to the photoinduced hot and trapped holes. The hole trapping process occurs on a subpicosecond time scale, followed by carrier recombination (~100 ps). The nature of the hole trapping sites, revealed by XTA, is characteristic of S or organic ligands on the surface of CuS NDs. These results not only suggest the possibility to control the hole dynamics by tuning the surface chemistry of CuS but also represent the first time observation of hole dynamics in semiconductor nanocrystals using XTA