Symbolic computation of conservation laws for nonlinear partial differential equations in multiple space dimensions

A method for symbolically computing conservation laws of nonlinear partial differential equations (PDEs) in multiple space dimensions is presented in the language of variational calculus and linear algebra. The steps of the method are illustrated using the Zakharov-Kuznetsov and Kadomtsev-Petviashvili equations as examples. The method is algorithmic and has been implemented in Mathematica. The software package, ConservationLawsMD.m, can be used to symbolically compute and test conservation laws for polynomial PDEs that can be written as nonlinear evolution equations. The code ConservationLawsMD.m has been applied to (2+1)-dimensional versions of the Sawada-Kotera, Camassa-Holm, and Gardner equations, and the multi-dimensional Khokhlov-Zabolotskaya equation.Comment: 26 pages. Paper will appear in Journal of Symbolic Computation (2011). Presented at the Special Session on Geometric Flows, Moving Frames and Integrable Systems, 2010 Spring Central Sectional Meeting of the American Mathematical Society, Macalester College, St. Paul, Minnesota, April 10, 201

Black Hole Scattering from Monodromy

We study scattering coefficients in black hole spacetimes using analytic properties of complexified wave equations. For a concrete example, we analyze the singularities of the Teukolsky equation and relate the corresponding monodromies to scattering data. These techniques, valid in full generality, provide insights into complex-analytic properties of greybody factors and quasinormal modes. This leads to new perturbative and numerical methods which are in good agreement with previous results.Comment: 28 pages + appendices, 2 figures. For Mathematica calculation of Stokes multipliers, download "StokesNotebook" from https://sites.google.com/site/justblackholes/techy-zon

Specifications-Based Grading Reduces Anxiety for Students of Ordinary Differential Equations

Specifications-based grading (SBG) is an assessment scheme in which student grades are based on demonstrated understanding of known specifications which are tied to course learning outcomes. Typically with SBG, students are given multiple opportunities to demonstrate such understanding. In undergraduate-level introductory ordinary differential equations courses at two institutions, SBG has been found to markedly decrease students’ self-reported anxiety related to the course as compared to traditionally graded courses

Fast computation of power series solutions of systems of differential equations

We propose new algorithms for the computation of the first N terms of a vector (resp. a basis) of power series solutions of a linear system of differential equations at an ordinary point, using a number of arithmetic operations which is quasi-linear with respect to N. Similar results are also given in the non-linear case. This extends previous results obtained by Brent and Kung for scalar differential equations of order one and two

Classification of polynomial integrable systems of mixed scalar and vector evolution equations. I

We perform a classification of integrable systems of mixed scalar and vector evolution equations with respect to higher symmetries. We consider polynomial systems that are homogeneous under a suitable weighting of variables. This paper deals with the KdV weighting, the Burgers (or potential KdV or modified KdV) weighting, the Ibragimov-Shabat weighting and two unfamiliar weightings. The case of other weightings will be studied in a subsequent paper. Making an ansatz for undetermined coefficients and using a computer package for solving bilinear algebraic systems, we give the complete lists of 2nd order systems with a 3rd order or a 4th order symmetry and 3rd order systems with a 5th order symmetry. For all but a few systems in the lists, we show that the system (or, at least a subsystem of it) admits either a Lax representation or a linearizing transformation. A thorough comparison with recent work of Foursov and Olver is made.Comment: 60 pages, 6 tables; added one remark in section 4.2.17 (p.33) plus several minor changes, to appear in J.Phys.

Minimal area submanifolds in AdS x compact

We describe the asymptotic behavior of minimal area submanifolds in product spacetimes of an asymptotically hyperbolic space times a compact internal manifold. In particular, we find that unlike the case of a minimal area submanifold just in an asymptotically hyperbolic space, the internal part of the boundary submanifold is constrained to be itself a minimal area submanifold. For applications to holography, this tells us what are the allowed "flavor branes" that can be added to a holographic field theory. We also give a compact geometric expression for the spectrum of operator dimensions associated with the slipping modes of the submanifold in the internal space. We illustrate our results with several examples, including some that haven't appeared in the literature before.Comment: 24 pages, no figure