Global well-posedness for a coupled modified KdV system

We prove the sharp global well-posedness result for the initial value problem (IVP) associated to the system of the modified Korteweg-de Vries (mKdV) equation. For the single mKdV equation such result has been obtained by using Mirura’s Transform that takes the KdV equation to the mKdV equation [8]. We do not know the existence of Miura’s Transform that takes a KdV system to the system we are considering. To overcome this difficulty we developed a new proof of the sharp global well-posedness result for the single mKdV equation without using Miura’s Transform. We could successfully apply this technique in the case of the mKdV system to obtain the desired result.     

Criterion of periodicity of solutions of a certain differential equation with a periodic coefficient

Summary In this paper the differential equation (1) y″=q(t)y is considered where q(t) is a real continuous function with period π. There is proved a necessary and sufficient condition for the stability of the trivial solution of Equation (1) when the zeros of the characteristic equation λ² - Aλ+1=0, coincide. Moreover, there is shown the construction of all Equations (1) admitting only periodic or half-periodic solutions with period π.     

Existence of a pair of positive solutions for a class of nonlinear eigenvalue problems

In this paper, we consider a nonlinear elliptic equation driven by the p-Laplacian and with a parameter λ > 0. Using a combination of variational and degree theoretic methods, we show that there exists λ^* > 0 such that, if λ > λ^*, then the problem has two positive smooth solutions. Our result extends earlier ones by Rabinowitz (semilinear equations) and Guo (nonlinear equations).      

Mathieu functions and coulomb spheroidal functions in the electrostatic probe theory

A spherical probe placed in a slowly moving collisional plasma with a large Debye length λ_D → ∞ is considered. The partial differential equation describing the electron concentration around the probe is reduced to two ordinary differential equations, namely, to the equation for Coulomb spheroidal functions and Mathieu’s modified equation with the parameter a of the latter related to the eigenvalue λ of the former by the relation a = λ + 1/4. It is shown that the solutions of Mathieu’s equation are Mathieu functions of half-integer order, which are expressed as series in terms of spherical Bessel functions and series of products of Bessel functions. These Mathieu functions are numerically constructed for Mathieu’s modified and usual equations.     

A two-space dimensional semilinear heat equation perturbed by (Gaussian) white noise

A two-space dimensional heat equation perturbed by a white noise in a bounded volume is considered. The equation is perturbed by a non-linearity of the type λ : f(AU) :, where :: means Wick (re)ordering with respect to the free solution;λ, A are small parameters, U denotes a solution, f is the Fourier transform of a complex measure with compact support. Existence and uniqueness of the solution in a class of Colombeau-Oberguggenberger generalized functions is proven. An explicit construction of the solution is given and it is shown that each term of the expansion in a power series in λ is associated with an L ²-valued measure when A is a small enough. Received: 20 July 1997 / Revised version: 1 February 2001 / Published online: 9 October 2001     

The initial value problem for cohomogeneity one Einstein metrics

The PDE Ric(g) = λ · g for a Riemannian Einstein metric g on a smooth manifold M becomes an ODE if we require g to be invariant under a Lie group G acting properly on M with principal orbits of codimension one. A singular orbit of the G-action gives a singularity of this ODE. Generically, an equation with such type of singularity has no smooth solution at the singularity. However, in our case, the very geometric nature of the equation makes it solvable. More precisely, we obtain a smooth G-invariant Einstein metric (with any Einstein constant λ) in a tubular neighbourhood around a singular orbit Q ⊂ M for any prescribed G-invariant metric g_Q and second fundamental form L_Q on Q, provided that the following technical condition is satisfied (which is very often the case): the representations of the principal isotropy group on the tangent and the normal space of the singular orbit Q have no common sub-representations. This Einstein metric is not uniquely determined by the initial data g_Q and L_Q; in fact, one may prescribe initial derivatives of higher degree, and examples show that this degree can be arbitrarily high. The proof involves a blend of ODE techniques and representation theory of the principal and singular isotropy groups.     

Lie jets and symmetries of prolongations of geometric objects

The Lie jet L _θ λ of a field of geometric objects λ on a smooth manifold M with respect to a field θ of Weil A-velocities is a generalization of the Lie derivative L _v λ of a field λ with respect to a vector field v. In this paper, Lie jets L _θ λ are applied to the study of A-smooth diffeomorphisms on a Weil bundle T ^A M of a smooth manifold M, which are symmetries of prolongations of geometric objects from M to T ^A M. It is shown that vanishing of a Lie jet L _θ λ is a necessary and sufficient condition for the prolongation λ ^A of a field of geometric objects λ to be invariant with respect to the transformation of the Weil bundle T ^A M induced by the field θ. The case of symmetries of prolongations of fields of geometric objects to the second-order tangent bundle T ² M are considered in more detail.     

To solving problems of algebra for two-parameter matrices. III

The paper continues the series of papers devoted to surveying and developing methods for solving algebraic problems for two-parameter polynomial and rational matrices of general form. It considers linearization methods, which allow one to reduce the problem of solving an equation F(λ, μ)x = 0 with a polynomial two-parameter matrix F(λ, μ) to solving an equation of the form D(λ, μ)y = 0, where D(λ, μ) = A(μ)-λB(μ) is a pencil of polynomial matrices. Consistent pencils and their application to solving spectral problems for the matrix F(λ, μ) are discussed. The notion of reducing subspace is generalized to the case of a pencil of polynomial matrices. An algorithm for transforming a general pencil of polynomial matrices to a quasitriangular pencil is suggested. For a pencil with multiple eigenvalues, algorithms for computing the Jordan chains of vectors are developed. Bibliography: 8 titles. Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 359, 2008, pp. 166–207.     

Penalty Approach to the HJB Equation Arising in European Stock Option Pricing with Proportional Transaction Costs

We present a novel penalty approach to the Hamilton-Jacobi-Bellman (HJB) equation arising from the valuation of European options with proportional transaction costs. We first approximate the HJB equation by a quasilinear 2nd-order partial differential equation containing two linear penalty terms with penalty parameters λ ₁ and λ ₂ respectively. Then, we show that there exists a unique viscosity solution to the penalized equation. Finally, we prove that, when both λ ₁ and λ ₂ approach infinity, the viscosity solution to the penalized equation converges to that of the corresponding original HJB equation.     

Risk Sensitive Control of Diffusions with Small Running Cost

Infinite horizon risk-sensitive control of diffusions is analyzed under a stability condition coupled with a bound on the running cost. It is shown that the corresponding Hamilton-Jacobi-Bellman equation has a solution (w(⋅),λ ^∗) where the scalar λ ^∗ is in fact the optimal cost. This also leads to an existence result for optimal controls.     

Motion of curves and solutions of two multi-component mKdV equations

Two classes of multi-component mKdV equations have been shown to be integrable. One class called the multi-component geometric mKdV equation is exactly the system for curvatures of curves when the motion of the curves is governed by the mKdV flow. In this paper, exact solutions including solitary wave solutions of the two- and three-component mKdV equations are obtained, the symmetry reductions of the two-component geometric mKdV equation to ODE systems corresponding to it’s Lie point symmetry groups are also given. Curves and their behavior corresponding to solitary wave solutions of the two-component geometric mKdV equation are presented.     

Exceptional Modules for Tubular Canonical Algebras

Let Λ be a tubular canonical algebra of quiver type over a field. We show that each exceptional Λ-module can be exhibited by matrices involving as coefficients 0, 1 and –1 if Λ is of type (3,3,3), (2,4,4) or (2,3,6) and by matrices involving as coefficients 0, 1, –1, λ, –λ and λ–1 if Λ is of type (2,2,2,2) and defined by a parameter λ. Presented by Claus M. Ringel.     

Optimization problems with algebraic solutions: Quadratic fractional programs and ratio games

A mathematical program with a rational objective function may have irrational algebraic solutions even when the data are integral. We suggest that for such problems the optimal solution will be represented as follows: If λ* denotes the optimal value there will be given an intervalI and a polynomialP(λ) such thatI contains λ* and λ* is the unique root ofP(λ) inI. It is shown that with this representation the solutions to convex quadratic fractional programs and ratio games can be obtained in polynomial time.     

Extremal barriers on cones with Phragmèn-Lindelöf theorems and other applications

Summary We give a detailed analysis of supersolutions of form rλf(θ) for the classL _α of uniformly elliptic operators in nondivergence form with ellipticity constant α. Fundamental to the analysis is the extremal supersolution rλFλ, α(θ) for each real λ, which is itself a solution of a certain extremal elliptic equation and which remains positive on a cone of wider aperature than any other supersolution of this form. These supersolutions are employed as “barriers” to yeild Phragmen-Lindel?f theorems (λ>0) for unbounded domains contained in a cone, and H?lder continuity (λ>0) and removeable singularity (λ<0) results at boundary points with an exterior cone property. In each case the growth condition 0(r_λ) involved, depending on the aperature of the cone, is best possible. Similar results carry through for operators with singular lower order terms and possibily positive zero order coefficient. This work was partially supported by AFOSR grant number 553–64. Entrata in Redazione il 29 gennaio 1971.     

On the asymptotics of the eigenvalue counting function for random recursive Sierpinski gaskets

We consider natural Laplace operators on random recursive affine nested fractals based on the Sierpinski gasket and prove an analogue of Weyl’s classical result on their eigenvalue asymptotics. The eigenvalue counting function N(λ) is shown to be of order λ ^ds/2 as λ→∞ where we can explicitly compute the spectral dimension d _s . Moreover the limit N(λ) λ ^−ds/2 will typically exist and can be expressed as a deterministic constant multiplied by a random variable. This random variable is a power of the limiting random variable in a suitable general branching process and has an interpretation as the volume of the fractal. Received: 22 January 1999 / Revised version: 2 September 1999 /?Published online: 30 March 2000     

New compacton-like and solitary patterns-like solutions to nonlinear wave equations with linear dispersion terms

It has been shown that many fully nonlinear wave equations with nonlinear dispersion terms possess compacton solutions and solitary patterns solutions. In this paper, with the aid of Maple, the mKdV equation, the equation with a source term, the five order KdV-like equation and the KdV–mKdV equation are investigated using some new, generalized transformations. As a consequence, it is shown that these equations with linear dispersion terms admit new compacton-like solutions and solitary patterns-like solutions. These transformations can be also extended to other nonlinear wave equations with nonlinear dispersion terms to seek new compacton-like solutions and solitary patterns-like solutions.     

The multi-component second modified Korteweg–de Vries equation and its two distinct integrable coupling types

A new isospectral problem is designed and the multi-component second mKdV equation is worked out from it. It follows that two distinct types of integrable couplings of the multi-component second mKdV equation are obtained by constructing two types of new loop algebras. As its reduction, two distinct types of integrable couplings of the multi-component KdV equation, the multi-component mKdV equation and the multi-component KdV–mKdV equation are presented.     

When is the union of two unit intervals a self-similar set satisfying the open set condition?

Let U _λ be the union of two unit intervals with gap λ. We show that U _λ is a self-similar set satisfying the open set condition if and only if U _λ can tile an interval by finitely many of its affine copies (admitting different dilations). Furthermore, each such λ can be characterized as the spectrum of an irreducible double word which represents a tiling pattern. Some further considerations of the set of all such λ’s, as well as the corresponding tiling patterns, are given. The first author was partially supported by the RGC grant and the direct grant in CUHK, Fok Ying Tong Education Foundation and NSFC (10571100). The second author was partially supported by NSFC (70371074) and NFSC (10571104).     

A Note on the Lie Symmetry Analysis and Exact Solutions for the Extended mKdV Equation

We discuss the recent paper by Liu and Li (Lie symmetry analysis and exact solutions for the extended mKdV equation, Acta Appl. Math. (2010), 109:1107?C1119) and illustrate that so called the ??extended mKdV equation?? by authors can be reduced to the usual form of the modified Korteweg?Cde Vries equation. We also correct some other statements by authors with respect to exact solutions of solutions for the ??extended mKdV equation??.     

Determination of the unknown sources in the heat-conduction equation

The article considers the determination of the source F(x,t) in the heat-conduction equation using additional information about the solution. The source F(x,t) is represented in the special form F(x,t)=f₁(x)exp(-λ₁t)+f₂(x)exp(-λ₂t) where λ₁,₂ are known positive constants. The additional information about the solution of the inverse problem is provided by the solution of the heat-conduction equation at a number of fixed points in space. Uniqueness of the solution of this inverse problem is investigated. Translated from Obratnye Zadachi Estestvoznaniya, Published by Moscow University, Moscow, 1997, pp. 18–22.