Structure of Equations Solvable by the Inverse Scattering Transform for the Schrödinger Operator

We propose a new approach for deriving nonlinear evolution equations solvable by the inverse scattering transform. The starting point of this approach is consideration of the evolution equations for the scattering data generated by solutions of an arbitrary nonlinear evolution equation that rapidly decrease as x±. Using this approach, we find all nonlinear evolution equations whose integration reduces to investigation of the scattering-data evolution equations that are differential equations (in either ordinary or partial derivatives). In this case, the evolution equations for the scattering data themselves are linear and moreover solvable in the finite form.     

Time decay for nonlinear wave equations in two space dimensions

The Cauchy Problem for the equation u_tt–u+|u|^p–1u=0 (x², t>0, >1) is studied. Smooth Cauchy data is prescribed, and no smallness condition is imposed. For >5, it is shown that the maximum amplitude of such a wave decays at the expected rate t^–1/2 as t. For 1+8<5, the maximum amplitude still decays, but at a slower rate. These results are then used to demonstrate the existence of the scattering operator when >_o, where _o is the root of the cubic equation ³-2²-7-8=0; thus _o4.15.Alfred P. Sloan Research Fellow     

On summability of subsequences of partial sums of trigonometric Fourier series

n- (n1) fL ^p ([–, ] ⁿ ),=1 = (L C) . , , f([–, ] ⁿ ).     

On Fourier multipliers between Besov spaces with 0 <p o ≤ min{1,p 1}

, 0<0 min{1,p1}.     

On the solvability of a complete second-order differential equation in Banach space

The paper deals with the Cauchy problem for a complete second-order differential equation with unbounded operator coefficientsu+A(t)u+B(t)u=f, u(0)=u0, u(0)=u ₁ . By using the commutant method, we construct a coercive solution of this problem in Holder space in the case where the operatorB is as strong as the operator A².Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 45, No. 10, pp. 1449–1454, October, 1993.     

On cubature formulas on the basis of lattices

Q (.. , L). Q . P(S_r(2)) — 2 (S _r(2) (r — ). , M(P(S _r(m=sup{t(·)t(·)₁:t P(S _r(2)),t 0}. , /4+(1)M(P(S _r(2)))/r ²15/17+(1)(r+). (Q), Q L.     

Stationary Solutions of the Fractional Kinetic Equation with a Symmetric Power-Law Potential

The properties of stationary solutions of the one-dimensional fractional Einstein--Smoluchowski equation with a potential of the form x ^2m+2, m=1,2,..., and of the Riesz spatial fractional derivative of order , 12, are studied analytically and numerically. We show that for 1<2, the stationary distribution functions have power-law asymptotic approximations decreasing as x ^–(+2m+1) for large values of the argument. We also show that these distributions are bimodal.     

Holomorphe Funktionen auf Gebieten über Banach-Räumen zu vorgegebenen Konvergenzradien

Given a function: ₊ on a domain spread over an infinite dimensional complex Banach space E with a Schauder basis such that -log is plurisubharmonic and d (d denotes the boundary distance on ) one can find a holomorphic function f: with _f, where _f is the radius of convergence of f. If, in addition, is locally Lipschitz continuous with constant 1, f can be chosen so that (3M)^–1 _f, where M is the basis constant of E. In the particular case of E= ₁ there are holomorphic functions f on with= _f.     

The Inverse Problem of the Calculus of Variations for Sixth- and Eighth-order Scalar Ordinary Differential Equations

On the equation manifold of the 2nth-order scalar ordinary differential equation, n3,

Orthogonal systems invariant under an automorphism

, , - ( ) . , - , ( ) .     

Exact error bound of approximation by interpolating splines inL-metric on the classesW p r (1≦p

Корнейчук  Н. П. 《Analysis Mathematica》1977,3(2):109-117

Radon—Nikodym theorems for multimeasures and transition multimeasures

— . , — . , .

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Research supported by N. S. F. Grant DMS-8802688.     

Investigation of a problem for the Laplace equation with a boundary condition of a special form in a plane angle

The explicit solution of the equation u=f in a plane infinite angle, satisfying on one side of the angle a Neumann condition and on the other one the condition u/n + hu/r + u= (/r is the tangential derivative, C, 0), is constructed and estimated in weighted Sobolev spaces. The obtained estimates are sharp with respect to the differential order and are uniform with respect to . The construction of the solution reduces to the investigation of a finite difference equation in the complex plane, arising after a Mellin transform.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova Akademii Nauk SSSR, Vol. 182, pp. 149–167, 1990.     

Замечание О Механичецкой Квадратуре

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Eine Existenzaussage für asymptotisch lineare Störungen eines Fredholmoperators mit Index O

Let X, Y be Banach spaces, L: X Dom(L)Y a linear Fredholm operator with Fredholm index zero, and T a asymptotic linear k-set contraction with k[0, 1(L)), where 1(L) denotes the lower bound of L with respect to the set-measure of noncampactness. Using coincidence degree we deduce a Fredholm alternative for the equation Lx=Tx+y (yY), which involves the results of [11] and [17] in the case X=Y and L=Id. Applications are given for a functional differential equation of neutral type and for a boundary value problem of a second order differential equation.     

Linearization of boundary eigenvalue problems

It is shown that certain eigenvalue problems for ordinary differential operators with boundary conditions depending holomorphically on the eigenvalue parameter can be linearized by making use of the theory of operator colligations. As examples, first order systems with boundary conditions depending polynomially on and Sturm-Liouville problems with -holomorphic boundary conditions are considered.     

On Sets of Planes in Projective Spaces Intersecting Mutually in One Point

Let be a projective space. In this paper we consider sets of planes of such that any two planes of intersect in exactly one point. Our investigation will lead to a classification of these sets in most cases. There are the following two main results:- If is a set of planes of a projective space intersecting mutually in one point, then the set of intersection points spans a subspace of dimension 6. There are up to isomorphism only three sets where this dimension is 6. These sets are related to the Fano plane.- If is a set of planes of PG(d,q) intersecting mutually in one point, and if q3, 3(q²+q+1), then is either contained in a Klein quadric in PG(5,q), or is a dual partial spread in PG(4,q), or all elements of pass through a common point.     

Embeddings of Lorentz—Marcinkiewicz spaces with mixed norms

X(Y) f -:X(Y)={fM(×): f_{X(Y)=f(x,.)YX<} . =(0, ), M (×) — , ×, X, Y, Z— . X(Y) Z(×).