On the exact WKB analysis of operators admitting infinitely many phases

To analyze differential operators whose WKB solutions admit infinitely many phases, we introduce a class of differential operators of WKB type and analyze their exact WKB theoretic structure near their turning points. Our analysis makes full use of techniques and ideas in microlocal analysis; we use a quantized contact transformation to construct a WKB solution of a differential equation of WKB type, and we use a Späth-type division theorem for a differential operator of WKB type to study its structure near turning points. As an application, we show a connection formula for WKB solutions near a simple turning point.     

A generalization of Riemann's method for partial differential equations

Summary A bilinear divergence identity is obtained, which differs from the usualLagrange divergence identity employed byRiemann. In the case of two independent variables, this new identity is used to unify the treatment ofCauchy's problem for hyperbolic equations, the initial value problem for parabolic equations, and theDirichlet problem for elliptic equations. This research was supported in whole or in part by the United States Air Force under Contract N^o. AF18(600)-573 monitored by the Office of Scientific Research, Air Research and Development Command.     

A class of relations among multiple zeta values

We prove a new class of relations among multiple zeta values (MZV's) which contains Ohno's relation. We also give the formula for the maximal number of independent MZV's of fixed weight, under our new relations. To derive our formula for MZV's, we consider the Newton series whose values at non-negative integers are finite multiple harmonic sums.     

Liouville-Green asymptotic approximation for a class of matrix differential equations and semi-discretized partial differential equations

A Liouville-Green (or WKB) asymptotic approximation theory is developed for the class of linear second-order matrix differential equations Y^″=[f(t)A+G(t)]Y on [a,+∞), where A and G(t) are matrices and f(t) is scalar. This includes the case of an “asymptotically constant” (not necessarily diagonalizable) coefficient A (when f(t)≡1). An explicit representation for a basis of the right-module of solutions is given, and precise computable bounds for the error terms are provided. The double asymptotic nature with respect to both t and some parameter entering the matrix coefficient is also shown. Several examples, some concerning semi-discretized wave and convection-diffusion equations, are given.     

Multiple Dedekind zeta functions and evaluations of extended multiple zeta values

We define the number field analog of the zeta function of d-complex variables studied by Zagier in (First European Congress of Mathematics, vol. II (Paris, 1992), Progress in Mathematics, vol. 120, Birkhauser, Basel, 1994, pp. 497-512). We prove that in certain cases this function has a meromorphic continuation to C^d, and we identify the linear subvarieties comprising its singularities. We use our approach to meromorphic continuation to prove that there exist infinitely many values of these functions at regular points in their extended domains which can be expressed as a rational linear combination of values of the Dedekind zeta function.     

A nonlinear analogue of the WKB method and the method of the averaged Lagrangian. II

This paper is a direct continuation of the author's previous paper. One considers questions concerning the uniqueness and the solvability of equations of the higher approximations of the WKB method and the relation between these questions and the properties of the averaged Lagrangian.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 128, pp. 95–101, 1983.     

A method for constructing solutions to linear systems of partial differential equations

We obtain some conditions of solvability in Sobolev spaces for the systems of linear partial differential equations and deduce the corresponding formulas for solutions to these systems. The solutions are given as the sum of the series whose terms are the iterations of some pseudodifferential operators constructed explicitly.     

Application of rational expansion method for stochastic differential equations

By means of the Hermite transformation, a new general ansätz and the symbolic computation system Maple, we apply the Riccati equation rational expansion method [24] to uniformly construct a series of stochastic non-traveling wave solutions for stochastic differential equations. To illustrate the effectiveness of our method, we take the stochastic mKdV equation as an example, and successfully construct some new and more general solutions. The method can also be applied to solve other nonlinear stochastic differential equation or equations.     

一类含Hysteresis微分子算子的反应扩散方程

In this paper, the classical and weak derivatives with respect to spatial variable of a class of hysteresis functional are discussed. Some conclusions about solutions of a class of reaction-diffusion equations with hysteresis differential operator are given.     

The WKB method and differential consequences of the Riccati equation

A generalized WKB method based on the use of the differential consequences of the Riccati equation is presented. The method combines the simplicity of the traditional WKB method and the universality of the Maslov method: in the case of a smooth potential with classical turning points in a bounded space interval, the leading term of the expansion is found as a root of an algebraic equation and provides a regular approximate solution in the whole domain of the potential; we can increase the accuracy of this solution by taking new differential consequences into account.     

Duality and double shuffle relations of multiple zeta values

We prove that certain families of duality relations of the multiple zeta values (MZV's) are consequences of the extended double shuffle relations (EDSR's), thereby proving a part of the conjecture that the EDSR's give all linear relations of the MZV's.     

Exact WKB analysis of a Schrödinger equation with a merging triplet of two simple poles and one simple turning point,II — Its relevance to the Mathieu equation and the Legendre equation

We develop the exact WKB analysis of an M2P1T (merging two simple poles and one simple turning point) Schrödinger equation. In Part II, using a WKB-theoretic transformation to the algebraic Mathieu equation constructed in Part I, we calculate the alien derivative of its Borel transformed WKB solutions at each fixed singular point relevant to the simple poles through the analysis of Borel transformed WKB solutions of the Legendre equations. In the course of the calculation of the alien derivative we make full use of microdifferential operators whose symbols are given by the infinite series that appear in the coefficients of the algebraic Mathieu equation and the Legendre equation.     

A rational generalization of Fan’s method and its application to generalized shallow water wave equations

In this paper we employ a rational expansion to generalize Fan’s method for exact travelling wave solutions for nonlinear partial differential equations (PDEs). To verify the reliability of the proposed method, the generalized shallow water wave (GSWW) equation has been investigated as an example. Kinds of new exact travelling wave solutions of a rational form have been obtained. This indicates that the proposed method provides a more general result for exact solution of nonlinear equations.     

Differential Algebraic Birkhoff Decomposition and the renormalization of multiple zeta values

In the Hopf algebra approach of Connes and Kreimer on renormalization of quantum field theory, the renormalization process is viewed as a special case of the Algebraic Birkhoff Decomposition. We give a differential algebra variation of this decomposition and apply this to the study of multiple zeta values.     

A generalization of Aubry–Mather theory to partial differential equations and pseudo-differential equations

We discuss an Aubry–Mather-type theory for solutions of non-linear, possibly degenerate, elliptic PDEs and other pseudo-differential operators.We show that for certain PDEs and ΨDEs with periodic coefficients and a variational structure it is possible to find quasi-periodic solutions for all frequencies. This results also hold under a generalized definition of periodicity that makes it possible to consider problems in covers of several manifolds, including manifolds with non-commutative fundamental groups.An abstract result will be provided, from which an Aubry–Mather-type theory for concrete models will be derived.     

Asymptotic behavior of the singular values of a generalization of the operator fractional integration

In this paper we find the first term in the asymptotics of singular values of the generalized fractional integration operator.     

A semiexplicit method for numerical solution of functional differential algebraic equations

We study systems with delay effect that contain additional algebraic relations. We propose semiexplicit numerical methods of the Rosenbrock type. We prove the solvability of equations of a numerical model and estimate the order of the global error. The chosen parameters provide the third order of the error.