Lie symmetry analysis of a class of time fractional nonlinear evolution systems

We study a class of nonlinear evolution systems of time fractional partial differential equations using Lie symmetry analysis. We obtain not only infinitesimal symmetries but also a complete group classification and a classification of group invariant solutions of this class of systems. We find that the class of systems of differential equations studied is naturally divided into two cases on the basis of the type of a function that they contain. In each case, the dimension of the Lie algebra generated by the infinitesimal symmetries is greater than 2, and for this reason we present the structures and one-dimensional optimal systems of these Lie algebras. The reduced systems corresponding to the optimal systems are also obtained. Explicit group invariant solutions are found for particular cases.Comment: 14 pages. To appear in Applied Mathematics and Computatio

Fractional Branes on a Non-compact Orbifold

Fractional branes on the non-compact orbifold \C^3/\Z_5 are studied. First, the boundary state description of the fractional branes are obtained. The open-string Witten index calculated using these states reproduces the adjacency matrix of the quiver of $\Z_5$. Then, using the toric crepant resolution of the orbifold \C^3/\Z_5 and invoking the local mirror principle, B-type branes wrapped on the holomorphic cycles of the resolution are studied. The boundary states corresponding to the five fractional branes are identified as bound states of BPS D-branes wrapping the 0-, 2- and 4-cycles in the exceptional divisor of the resolution of \C^3/\Z_5.Comment: Latex2e, 25 pages, typos corrected, minor modifications, version to appear in JHE

Stokes Phenomena and Quantum Integrability in Non-critical String/M Theory

We study Stokes phenomena of the k \times k isomonodromy systems with an arbitrary Poincar\'e index r, especially which correspond to the fractional-superstring (or parafermionic-string) multi-critical points (\hat p,\hat q)=(1,r-1) in the k-cut two-matrix models. Investigation of this system is important for the purpose of figuring out the non-critical version of M theory which was proposed to be the strong-coupling dual of fractional superstring theory as a two-matrix model with an infinite number of cuts. Surprisingly the multi-cut boundary-condition recursion equations have a universal form among the various multi-cut critical points, and this enables us to show explicit solutions of Stokes multipliers in quite wide classes of (k,r). Although these critical points almost break the intrinsic Z_k symmetry of the multi-cut two-matrix models, this feature makes manifest a connection between the multi-cut boundary-condition recursion equations and the structures of quantum integrable systems. In particular, it is uncovered that the Stokes multipliers satisfy multiple Hirota equations (i.e. multiple T-systems). Therefore our result provides a large extension of the ODE/IM correspondence to the general isomonodromy ODE systems endowed with the multi-cut boundary conditions. We also comment about a possibility that N=2 QFT of Cecotti-Vafa would be "topological series" in non-critical M theory equipped with a single quantum integrability.Comment: 43 pages, 3 figures; v2:references and comments added (footnote 24

Symmetry Analysis of Initial and Boundary Value Problems for Fractional Differential Equations in Caputo sense

In this work we study Lie symmetry analysis of initial and boundary value problems for partial differential equations (PDE) with Caputo fractional derivative. We give generalized definition and theorem for the symmetry method for PDE with Caputo fractional derivative, according to Bluman's definition and theorem for the symmetry analysis of PDE system. We investigate the symmetry analysis of initial and boundary value problem for fractional diffusion and the third order fractional partial differential equation (FPDE). Also we give some solutions

Fractional D1-Branes at Finite Temperature

The supergravity dual of $N$ regular and $M$ fractional D1-branes on the cone over the Einstein manifold $Q^{1,1,1}$ has a naked singularity in the infrared. The supergravity dual of $N$ regular and $M$ fractional D3-branes on the conifold also has such a singularity. Buchel suggested and Gubser et al. have shown that in the D3-brane case, the naked singularity is cloaked by a horizon at a sufficiently high temperature. In this paper we derive the system of second-order differential equations necessary to find such a solution for $Q^{1,1,1}$. We also find solutions to this system in perturbation theory that is valid when the Hawking temperature of the horizon is very high.Comment: 22 pages, v2: minor change