Multidimensional linearizable system of <Emphasis Type="Italic">n</Emphasis>-wave-type equations

We propose a linearizable version of a multidimensional system of n-wave-type nonlinear partial differential equations (PDEs). We derive this system using the spectral representation of its solution via a procedure similar to the dressing method for nonlinear PDEs integrable by the inverse scattering transform method. We show that the proposed system is completely integrable and construct a particular solution.     

Estimates for the widths of discrete function classes generated by a two-weight summation operator

Order estimates are obtained for the Kolmogorov, linear, and Gel’fand widths of the image of the unit ball of the space l_p under the action of a two-weight summation operator with weights of special kind. Some limit conditions on the parameters defining the weights are considered.     

Integrability Properties of A Supersymmetric Coupled Dispersionless Integrable System

We study the integrability aspects of an N=1 supersymmetric coupled dispersionless (SUSY-CD) integrable system in detail. We present a superfield Lax representation of the SUSY-CD system by writing its (3×3)-matrix superfield Lax pair and show that the zero-curvature condition corresponds to the SUSY-CD system. From the fermionic superfield Lax representation, we obtain a set of coupled superfield Riccati equations that we further use to obtain an infinite set of superfield conserved currents. We investigate the Darboux transformation of the SUSY-CD system and use it to obtain multisoliton solutions of the system.     

Wick Calculus for Noncommutative White Noise Corresponding to <Emphasis Type="Italic">q</Emphasis>-Deformed Commutation Relations

We derive the Wick calculus for test and generalized functionals of noncommutative white noise corresponding to q-deformed commutation relations with \(q\in (-1,1)\). We construct a Gel’fand triple centered at the q-deformed Fock space in which both the test, nuclear space and its dual space are algebras with respect to the addition and the Wick multiplication. Furthermore, we prove a Våge-type inequality for the Wick product on the dual space.     

On the determination of the potential function from given orbits

The paper deals with the problem of finding the field of force that generates a given (N ? 1)-parametric family of orbits for a mechanical system with N degrees of freedom. This problem is usually referred to as the inverse problem of dynamics. We study this problem in relation to the problems of celestial mechanics. We state and solve a generalization of the Dainelli and Joukovski problem and propose a new approach to solve the inverse Suslov’s problem. We apply the obtained results to generalize the theorem enunciated by Joukovski in 1890, solve the inverse Stäckel problem and solve the problem of constructing the potential-energy function U that is capable of generating a bi-parametric family of orbits for a particle in space. We determine the equations for the sought-for function U and show that on the basis of these equations we can define a system of two linear partial differential equations with respect to U which contains as a particular case the Szebehely equation. We solve completely a special case of the inverse dynamics problem of constructing U that generates a given family of conics known as Bertrand’s problem. At the end we establish the relation between Bertrand’s problem and the solutions to the Heun differential equation. We illustrate our results by several examples.     

Estimates for a spectral parameter in elliptic boundary value problems with discontinuous nonlinearities

Under study are the two classes of elliptic spectral problems with homogeneous Dirichlet conditions and discontinuous nonlinearities (the parameter occurs in the nonlinearity multiplicatively). In the former case the nonlinearity is nonnegative and vanishes for the values of the phase variable not exceeding some positive number c; it has linear growth at infinity in the phase variable u and the only discontinuity at u = c. We prove that for every spectral parameter greater than the minimal eigenvalue of the differential part of the equation with the homogeneous Dirichlet condition, the corresponding boundary value problem has a nontrivial strong solution. The corresponding free boundary in this case is of zero measure. A lower estimate for the spectral parameter is established as well. In the latter case the differential part of the equation is formally selfadjoint and the nonlinearity has sublinear growth at infinity. Some upper estimate for the spectral parameter is given in this case.     

Mixed problem for the wave equation with integrable potential in the case of two-point boundary conditions of distinct orders

We study a mixed problem for the wave equation with integrable potential and with two-point boundary conditions of distinct orders for the case in which the corresponding spectral problem may have multiple spectrum. Based on the resolvent approach in the Fourier method and the Krylov convergence acceleration trick for Fourier series, we obtain a classical solution u(x, t) of this problem under minimal constraints on the initial condition u(x, 0) = ?(x). We use the Carleson–Hunt theorem to prove the convergence almost everywhere of the formal solution series in the limit case of ?(x) ∈ L _p[0, 1], p > 1, and show that the formal solution is a generalized solution of the problem.     

Equivariant perturbation in Gomory and Johnson’s infinite group problem—III: foundations for the k-dimensional case with applications to $$ $$

We develop foundational tools for classifying the extreme valid functions for the k-dimensional infinite group problem. In particular, we present the general regular solution to Cauchy’s additive functional equation on restricted lower-dimensional convex domains. This provides a k-dimensional generalization of the so-called Interval Lemma, allowing us to deduce affine properties of the function from certain additivity relations. Next, we study the discrete geometry of additivity domains of piecewise linear functions, providing a framework for finite tests of minimality and extremality. We then give a theory of non-extremality certificates in the form of perturbation functions. We apply these tools in the context of minimal valid functions for the two-dimensional infinite group problem that are piecewise linear on a standard triangulation of the plane, under a regularity condition called diagonal constrainedness. We show that the extremality of a minimal valid function is equivalent to the extremality of its restriction to a certain finite two-dimensional group problem. This gives an algorithm for testing the extremality of a given minimal valid function.     

Linear FDEs in the frame of generalized ODEs: variation-of-constants formula

We present a variation-of-constants formula for functional differential equations of the form

$$\dot y = \mathcal{L}\left( t \right)y_t + f\left( {y_t,t} \right),\;y_{t_0}= \varphi $$

, where \(\mathcal{L}\) is a bounded linear operator and φ is a regulated function. Unlike the result by G. Shanholt (1972), where the functions involved are continuous, the novelty here is that the application t \(t \mapsto f\left( {y_t,t} \right)\) is Kurzweil integrable with t in an interval of ?, for each regulated function y. This means that t \(t \mapsto f\left( {y_t,t} \right)\) may admit not only many discontinuities, but it can also be highly oscillating and yet, we are able to obtain a variation-of-constants formula. Our main goal is achieved via theory of generalized ordinary differential equations introduced by J.Kurzweil (1957). As a matter of fact, we establish a variation-of-constants formula for general linear generalized ordinary differential equations in Banach spaces where the functions involved are Kurzweil integrable. We start by establishing a relation between the solutions of the Cauchy problem for a linear generalized ODE of type

$$\frac{{dx}}{{d\tau }} = D\left[ {A\left( t \right)x} \right],x\left( {{t_0}} \right) = \tilde x$$

and the solutions of the perturbed Cauchy problem

$$\frac{{dx}}{{d\tau }} = D\left[ {A\left( t \right)x + F\left( {x,t} \right)} \right],x\left( {{t_0}} \right) = \tilde x$$

Then we prove that there exists a one-to-one correspondence between a certain class of linear generalized ODE and the Cauchy problem for a linear functional differential equations of the form

$$\dot y = \mathcal{L}\left( t \right)y_t,\;y_{t_0} = \varphi$$

, where \(\mathcal{L}\) is a bounded linear operator and φ is a regulated function. The main result comes as a consequence of such results. Finally, because of the extent of generalized ODEs, we are also able to describe the variation-of-constants formula for both impulsive FDEs and measure neutral FDEs.

     

On Lie’s problem and differential invariants of ODEs <Emphasis Type="Italic">y</Emphasis>″ = <Emphasis Type="Italic">F</Emphasis>(<Emphasis Type="Italic">x</Emphasis>, <Emphasis Type="Italic">y</Emphasis>)

A point classification of ordinary differential equations of the form y″ = F(x, y) is considered. The algebra of differential invariants of the action of the point symmetry pseudogroup on the right-hand sides of equations of the form y″ = F(x, y) is calculated, and Lie’s problem on the point equivalence of such equations is solved.     

Solution of boundary value problems for surfaces of prescribed mean curvature <Emphasis Type="Italic">H</Emphasis> (<Emphasis Type="Italic">x,y, z</Emphasis>) with 1–1 central projection via the continuity method

When we consider surfaces of prescribed mean curvature H with a one-to-one orthogonal projection onto a plane, we have to study the nonparametric H-surface equation. Now the H-surfaces with a one-to-one central projection onto a plane lead to an interesting elliptic differential equation, which has been discovered for the case H = 0 already by T. Radó in 1932. We establish the uniqueness of the Dirichlet problem for this H-surface equation in central projection and develop an estimate for the maximal deviation of large H-surfaces from their boundary values, resembling an inequality by J. Serrin from 1969.We solve the Dirichlet problem for nonvanishing H with compact support via a nonlinear continuity method. Here we introduce conformal parameters into the surface and study the well-known H-surface system. Then we combine these investigations with a differential equation for its unit normal, which has been developed by the author for variable H in 1982. Furthermore, we construct large H-surfaces bounding extreme contours by an approximation.Here we only provide an overview on the relevant proofs; for the more detailed derivations of our results, we refer the readers to the author’s investigations in the Pacific Journal of Mathematics and the Milan Journal of Mathematics.     

The integrable case of Adler–van Moerbeke. Discriminant set and bifurcation diagram

The Adler–van Moerbeke integrable case of the Euler equations on the Lie algebra so(4) is investigated. For the L–A pair found by Reyman and Semenov-Tian-Shansky for this system, we explicitly present a spectral curve and construct the corresponding discriminant set. The singularities of the Adler–van Moerbeke integrable case and its bifurcation diagram are discussed. We explicitly describe singular points of rank 0, determine their types, and show that the momentum mapping takes them to self-intersection points of the real part of the discriminant set. In particular, the described structure of singularities of the Adler–van Moerbeke integrable case shows that it is topologically different from the other known integrable cases on so(4).     

Nekhoroshev estimates for commuting nearly integrable symplectomorphisms

In this paper, we prove the Nekhoroshev estimates for commuting nearly integrable symplectomorphisms. We show quantitatively how ? ^m symmetry improves the stability time. This result can be considered as a counterpart of Moser’s theorem [11] on simultaneous conjugation of commuting circle maps in the context of Nekhoroshev stability. We also discuss the possibility of Tits’ alternative for nearly integrable symplectomorphisms.     

A Method to Study the Cauchy Problem for a Singularly Perturbed Homogeneous Linear Differential Equation of Arbitrary Order

We construct a sequence converging to the solution to the Cauchy problem for a singularly perturbed linear homogeneous differential equation of any order. This sequence is asymptotic in the following sense: the distance (with respect to the norm of the space of continuous functions) between its nth element and the solution to the problem is proportional to the (n + 1)th power of the perturbation parameter.     

Spectrum transformation and conservation laws of lattice potential KdV equation

Many multi-dimensional consistent discrete systems have soliton solutions with nonzero backgrounds, which brings difficulty in the investigation of integrable characteristics. In this paper, we derive infinitely many conserved quantities for the lattice potential Korteweg-de Vries equation whose solutions have nonzero backgrounds. The derivation is based on the fact that the scattering data a(z) is independent of discrete space and time and the analytic property of Jost solutions of the discrete Schrödinger spectral problem. The obtained conserved densities are asymptotic to zero when |n| (or |m|) tends to infinity. To obtain these results, we reconstruct a discrete Riccati equation by using a conformal map which transforms the upper complex plane to the inside of unit circle. Series solution to the Riccati equation is constructed based on the analytic and asymptotic properties of Jost solutions.     

Evolution equations with Gel’fond-Leont’ev generalized differentiation operators: II

We establish the solvability of the Cauchy problem for evolution equations with Gel’fond-Leont’ev generalized differentiation operators in spaces of the type W as well as in spaces of generalized functions (analytic functionals) of the type W′.     

Applications of a few Lie algebras

Two isomorphic groups R ² andM are firstly constructed. Then we extend them into the differential manifold R ²ⁿ and n products of the group M for which four kinds of Lie algebras are obtained. By using these Lie algebras and the Tu scheme, integrable hierarchies of evolution equations along with multi-component potential functions can be generated, whose Hamiltonian structures can be worked out by the variational identity. As application illustrations, two integrable Hamiltonian hierarchies with 4 component potential functions are obtained, respectively, some new reduced equations are followed to present. Specially remark that the integrable hierarchies obtained by taking use of the approach presented in the paper are not integrable couplings. Finally, we generalize an equation obtained in the paper to introduce a general nonlinear integrable equation with variable coefficients whose bilinear form, B¨acklund transformation, Lax pair and infinite conserved laws are worked out, respectively, by taking use of the Bell polynomials.     

Convergence analysis of the spectral collocation methods for two-dimensional nonlinear weakly singular Volterra integral equations*

We apply Jacobi spectral collocation approximation to a two-dimensional nonlinear weakly singular Volterra integral equation with smooth solutions. Under reasonable assumptions on the nonlinearity, we carry out complete convergence analysis of the numerical approximation in the L^∞-norm and weighted L²-norm. The provided numerical examples show that the proposed spectral method enjoys spectral accuracy.     

A Riemann-Hilbert Approach to the Chen-Lee-Liu Equation on the Half Line

In this paper, the Fokas unified method is used to analyze the initial-boundary value for the Chen- Lee-Liu equation

$i{\partial _t}u + {\partial_{xx}u - i |u{|^2}{\partial _x}u = 0}$

on the half line (?∞, 0] with decaying initial value. Assuming that the solution u(x, t) exists, we show that it can be represented in terms of the solution of a matrix Riemann-Hilbert problem formulated in the plane of the complex spectral parameter λ. The jump matrix has explicit (x, t) dependence and is given in terms of the spectral functions {a(λ), b(λ)} and {A(λ), B(λ)}, which are obtained from the initial data u₀(x) = u(x, 0) and the boundary data g₀(t) = u(0, t), g₁(t) = u_x(0, t), respectively. The spectral functions are not independent, but satisfy a so-called global relation.

     

An Analytic Expression for the Distribution of the Generalized Shiryaev–Roberts Diffusion

We consider the quickest change-point detection problem where the aim is to detect the onset of a pre-specified drift in “live”-monitored standard Brownian motion; the change-point is assumed unknown (nonrandom). The topic of interest is the distribution of the Generalized Shryaev–Roberts (GSR) detection statistic set up to “sense” the presence of the drift. Specifically, we derive a closed-form formula for the transition probability density function (pdf) of the time-homogeneous Markov diffusion process generated by the GSR statistic when the Brownian motion under surveillance is “drift-free”, i.e., in the pre-change regime; the GSR statistic’s (deterministic) nonnegative headstart is assumed arbitrarily given. The transition pdf formula is found analytically, through direct solution of the respective Kolmogorov forward equation via the Fourier spectral method to achieve separation of the spacial and temporal variables. The obtained result generalizes the well-known formula for the (pre-change) stationary distribution of the GSR statistic: the latter’s stationary distribution is the temporal limit of the distribution sought in this work. To conclude, we exploit the obtained formula numerically and briefly study the pre-change behavior of the GSR statistic versus three factors: (a) drift-shift magnitude, (b) time, and (c) the GSR statistic’s headstart.