Bäcklund autotransformation for the equationu xt=eu−e−2u

The system of differential relations that arises in connection with the Bullough-Dodd-Zhiber-Shabat equationu _xt=e^u–e^–2u is considered. The consistency of this system is established, and it is shown that the system realizes a Bäcklund autotransformation for the equationu _xt=e^u–e^–2u. The associated three-dimensional dynamical systems, which are compatible on a two-dimensional invariant submanifold, are investigated, and a construction of their general solution, which gives the explicit form of the three-parameter soliton for the equationu _xt=e^u–e^–2u, is proposed.Bashkir State University. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 95, No. 1, pp. 146–159, April, 1993.     

Functions of class II

We consider the class II of operator functions f(z), meromorphic for ¦z¦1, which admit a representation in the form f(z)=f ₂ ^–1 (z)f₁(z), where f₁(z) is an operatorvalued holomorphic function and f₂(z) a scalar-valued bounded holomorphic function, such that the strong limits and coincide a.e. on the unit circle ¦¦=1. We prove that any function of class II can be represented as a block of some J-inner function of class II. We describe all such representations. The results found are applied to the question of realizations of functions of class II as transfer functions of linear systems.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta.im. V. A. Steklova AN SSSR, Vol. 135, pp. 5–30, 1984.     

On quasilinear Beltrami-type equations with degeneration

We consider the solvability problem for the equation $f_{\bar z} $ = v(z, f(z))f _z , where the function v(z,w) of two variables may be close to unity. Such equations are called quasilinear Beltrami-type equations with ellipticity degeneration. We prove that, under some rather general conditions on v(z,w), the above equation has a regular homeomorphic solution in the Sobolev classW _loc ^1,1 . Moreover, such solutions f satisfy the inclusion f ^?1 ∈ W _loc ^1,2 .     

Qp-valued jump processes associated with linear and nonlinear pseudo-differential equations

We consider stochastic differential equations in the space Q_p of p-adic numbers associated with nonlinear pseudo-differential equations and systems. We construct both scalar and matrix-valued multiplicative functionals of Q_p-valued Markov processes and derive probabilistic representations of solutions to the Cauchy problem for nonlinear pseudo-differential equations and systems which extend the Feynman-Kac formula. Bibliography: 12 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 351, 2007, pp. 5–37.     

Boundary-value problems for x-analytical functions with weighted boundary conditions

We consider boundary-value problems for x-analytical functions of a complex variable z = x + iy in a number of domains. Limit values with the weight (ln x)^–1 are given for the real part of the x-analytical function on the sections of the boundary that follow the imaginary axis, and simple limits are given for the real part of the x-analytical functions on the part of the boundary outside the imaginary axis. The apparatus of integral representations of x-analytical functions is applied to obtain a solution of the problem in quadratures.Kiev University. Translated from Vychislitel'naya i Prikladnaya Matematika, No. 74, pp. 3–11, 1992;     

Uniqueness theorem of solutions for stochastic differential equation in the plane

LetM={M _z, z ∈ R ₊ ² } be a continuous square integrable martingale andA={A _z, z ∈ R ₊ ² be a continuous adapted increasing process. Consider the following stochastic partial differential equations in the plane:dX _z=α(z, X_z)dM_z+β(z, X_z)dA_z, z∈R ₊ ² , X_z=Z_z, z∈∂R ₊ ² , whereR ₊ ² =[0, +∞)×[0,+∞) and ∂R ₊ ² is its boundary,Z is a continuous stochastic process on ∂R ₊ ² . We establish a new theorem on the pathwise uniqueness of solutions for the equation under a weaker condition than the Lipschitz one. The result concerning the one-parameter analogue of the problem we consider here is immediate (see [1, Theorem 3.2]). Unfortunately, the situation is much more complicated for two-parameter process and we believe that our result is the first one of its kind and is interesting in itself. We have proved the existence theorem for the equation in [2]. Supported by the National Science Foundation and the Postdoctoral Science Foundation of China     

On Equations Which Describe Pseudospherical Surfaces

Using the notion of a differential equation which describes an η-pseudospherical surface (η-p.s.s.), we give a characterization of the equations of type u_xt = F(u, u_x,…, ?^ku / ?x^k), k ≥ 2, with this property. We obtain a systematic procedure to determine a linear problem for which a given equation is the integrability condition. The equations of type u_xt = F(u, u_x) were characterized by Rabelo and Tenenblat in another paper. The theory is applied to several equations, some of which were not known to describe η-p.s.s.     

On the Distribution Modulo 1 of q-Additive Functions

We consider the asymptotic behavior of the distribution functions defined by F_N(z)=¹―_N{1≦ n ≦ N : f(n)≦ z (mod 1)} in the case when f is q-additive. We give necessary and sufficient conditions for a q-additive function to have a uniform distribution modulo 1 or to have a non-uniform distribution modulo 1. This revised version was published online in June 2006 with corrections to the Cover Date.     

On the sets of boundedness of solutions for a class of degenerate nonlinear elliptic fourth-order equations with <Emphasis Type="Italic">L</Emphasis><Superscript>1</Superscript>-data

In this article, we deal with a class of degenerate, nonlinear, elliptic fourth-order equations in divergence form with coefficients satisfying a strengthened ellipticity condition and right-hand sides of the class L ¹ depending on the unknown function. We consider the Dirichlet problem for equations of the given class and prove the existence of solutions of this problem bounded on the sets where the behavior of the data of the problem and the weighted functions involved is sufficiently regular. __________ Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 12, No. 4, pp. 99–112, 2006.     

A Greedy Subspace Method for Computing the ℒ∞-Norm

We consider the computation of the ℒ_∞-norm for a general class of ℒ_∞-functions and focus on the case where the function is represented in terms of large-scale matrix-valued factors. We propose a subspace projection method to obtain reduced approximations of this function by interpolation techniques. The ℒ_∞-norms are computed for the resulting reduced functions, then the subspaces are refined by means of the optimizer of the ℒ_∞-norm of the reduced function. In this way we obtain much better performance compared to existing methods. (© 2017 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Solution of two-dimensional boundary value problems corresponding to initial-boundary value problems of diffusion on a right cylinder

We consider boundary value problems for the differential equations Δ₂ u + B u = 0 with operator coefficients B corresponding to initial-boundary value problems for the diffusion equation Δ₃ u − pu = ∂ _t u (p > 0) on a right cylinder with inhomogeneous boundary conditions on the lateral surface of the cylinder with zero boundary conditions on the bases of the cylinder and with zero initial condition. For their solution, we derive specific boundary integral equations in which the space integration is performed only over the lateral surface of the cylinder and the kernels are expressed via the fundamental solution of the two-dimensional heat equation and the Green function of corresponding one-dimensional initial-boundary value problems of diffusion. We prove uniqueness theorems and obtain sufficient existence conditions for such solutions in the class of functions with continuous L ₂-norm.     

Super Riemann theta function periodic wave solutions and rational characteristics for a supersymmetric KdV-Burgers equation

Using a multidimensional super Riemann theta function, we propose two key theorems for explicitly constructing multiperiodic super Riemann theta function periodic wave solutions of supersymmetric equations in the superspace ℝ _Λ^N+1,M, which is a lucid and direct generalization of the super-Hirota-Riemann method. Once a supersymmetric equation is written in a bilinear form, its super Riemann theta function periodic wave solutions can be directly obtained by using our two theorems. As an application, we present a supersymmetric Korteweg-de Vries-Burgers equation. We study the limit procedure in detail and rigorously establish the asymptotic behavior of the multiperiodic waves and the relations between periodic wave solutions and soliton solutions. Moreover, we find that in contrast to the purely bosonic case, an interesting phenomenon occurs among the super Riemann theta function periodic waves in the presence of the Grassmann variable. The super Riemann theta function periodic waves are symmetric about the band but collapse along with the band. Furthermore, the results can be extended to the case N > 2; here, we only consider an existence condition for an N-periodic wave solution of a general supersymmetric equation.     

A note on the regularity criteria for the Navier-Stokes equations

We consider the regularity problem for 3D Navier-Stokes equations in a bounded domain with smooth boundary. A new sufficient condition which guarantees the regularity of weak solutions on the quotient ∇p/(1+|u|^δ₁+|∇u|^δ₂) for the Navier-Stokes equations is established.     

New approach to the solution of the classical sine-Gordon equation and its generalizations

We obtain new exact solutions U(x, y, z, t) of the three-dimensional sine-Gordon equation. The three-dimensional solutions depend on an arbitrary function F(α) whose argument is a function α(x, y, z, t). The ansatz α is found from an equation linear in (x, y, z, t) whose coefficients are arbitrary functions of α that should satisfy a system of algebraic equations. By this method, we solve the classical and a generalized sine-Gordon equation; the latter additionally contains first derivatives with respect to (x, y, z, t). We separately consider an equation that contains only the first derivative with respect to time. We present approaches to the solution of the sine-Gordon equation with variable amplitude. The considered methods for solving the sine-Gordon equation admit a natural generalization to the case of integration of the same types of equations in a space of arbitrarily many dimensions.     

Stabilization of statistics in wave and Klein-Gordon equations with mixing. Scattering theory for infinite energy solutions

We consider wave and Klein-Gordon equations in the whole space ?ⁿ with arbitraryn≥2. We assume initial data to be homogeneous random functions in ?ⁿ with zero expectation and finite mean density of energy. Moreover, we assume initial data fit mixing condition of Ibragimov-Linnik type. We consider the distributions of the random solution at the moment of timet. The main results mean the convergence of this distribution to some Gaussian measure ast→∞. This is a central limit theorem for wave and Klein-Gordon equations. The limit Gaussian measures are invariant measures for equations considered. Corresponding stationary random solutions are ergodic and mixing in time. The results are inspired by mathematical problems of statistical physics.     

Unique solvability of certain matrix partial differential equations

A class of matrix-valued functions is picked out, invariant relative to the operator, where t=(t₁, ..., t_n) are complex variables, x is a real parameter, A(x) is a matrix, {_i(x)} ₁ ⁿ =(A(x)). It is shown that the operator is normally solvable in the class picked out and a uniqueness theorem is proved for the solution of a nonstandard problem: the desired matrix-valued function Z(x, t) is known only at a point andz/x Ker ^*. Such problems arise naturally when developing the general theory of singular perturbations.Translated from Matematicheskie Zametki, Vol. 21, No. 4, pp. 525–530, April, 1977.     

A class of quasilinear parabolic equations with infinite delay and application to a problem of viscoelasticity

A semigroup approach to differential-delay equations is developed which reduces such equations to ordinary differential equations on a Banach space of histories and seems more suitable for certain partial integro-differential equations than the standard theory. The method is applied to prove a local-time existence theorem for equations of the form u_tt = g(u_xt, u_x^t)_x, where $\text{?g}\text{?u}{}_{\mathrm{xt}}\text{> 0}$. On a formal level, it is demonstrated that the stretching of filaments of viscoelastic liquids can be described by an equation of this form.     

Approximation by Faber-Laurent rational functions in the mean of functions of callsL p(Γ) with 1

Çavu§  A.  Israfilov  D. M. 《分析论及其应用》1995,11(1):105-118