On a direct procedure for the disclosure of Lax pairs and Bäcklund transformations

A direct and unifying scheme for the disclosure of bilinear Bäcklund transformations and linear Lax systems associated with soliton equations is presented. The scheme is based on a concept of scale invariance and on the use of a class of partitional polynomials: the binary Bell polynomials. The applicability of the procedure is tested on a variety of soliton equations.     

Towards second order Lax pairs to discrete Painlevé equations of first degree

We investigate the question of finding discrete Lax pairs for the six discrete Painlevé equations (Pn). The choice we make is to discretize the pairs of Garnier, once converted to matrix form.     

Lax pairs and Bäcklund transformations for a coupled Ramani equation and its related system

A coupled Ramani equation and its related system are proposed. By dependent variable transformation, they are transformed into bilinear equations. Lax pairs and Bäcklund transformations are presented for these two systems. Soliton solutions and rational solutions to the systems could be obtained.     

A simple characterization of the set ofμ-entropy pairs and applications

We present simple characterizations of the setsE _μ andE _X of measure entropy pairs and topological entropy pairs of a topological dynamical system (X, T) with invariant probability measureμ. This characterization is used to show that the set of (measure) entropy pairs of a product system coincides with the product of the sets of (measure) entropy pairs of the component systems; in particular it follows that the product of u.p.e. systems (topological K-systems) is also u.p.e. Another application is to show that the proximal relationP forms a residual subset of the setE _X . Finally an example of a minimal point distal dynamical system is constructed for whichE _X ∩(X ₀×X ₀)≠ , whereX ₀ is the denseG _δ subset of distal points inX.     

Continuity and equicontinuity of semigroups on norming dual pairs

We study continuity and equicontinuity of semigroups on norming dual pairs with respect to topologies defined in terms of the duality. In particular, we address the question whether continuity of a semigroup already implies (local/quasi) equicontinuity. We apply our results to transition semigroups and show that, under suitable hypothesis on E, every transition semigroup on C _b (E) which is continuous with respect to the strict topology β ₀ is automatically quasi-equicontinuous with respect to that topology. We also give several characterizations of β ₀-continuous semigroups on C _b (E) and provide a convenient condition for the transition semigroup of a Banach space valued Markov process to be β ₀-continuous.     

Stäckel transform of Lax equations

We construct a map between Lax equations for pairs of Liouville integrable Hamiltonian systems related by a multiparameter Stäckel transform. Using this map, we construct Lax representation for a wide class of separable systems by applying the multiparameter Stäckel transform to Lax equations of suitably chosen systems from a seed class. For a given separable system, we obtain in this way a set of nonequivalent Lax equations parameterized by an arbitrary function of the spectral parameter, as it is in the case of a related seed system.     

Lax–Hopf formula and Max-Plus properties of solutions to Hamilton–Jacobi equations

We state and prove a “Lax–Hopf formula” characterizing viable capture basins of targets investigated in viability theory and derive a “Max-Plus” morphism of capture basins with respect to the target. Capture basins are used to define “viability solutions” to Hamilton–Jacobi equations satisfying “trajectory conditions” (initial, boundary or Lagrangian conditions). The Max-Plus morphism property of Lax–Hopf formula implies the fact that the solution associated with inf-convolution of trajectory conditions is the inf-convolution of the solutions for each trajectory condition. For instance, Lipschitz regularization or decreasing envelopes of trajectory condition imply the Lipschitz regulation or decreasing envelopes of the solutions.     

A generalization of the Beurling—Lax theorem

We obtain conditions for the completeness of the system {G(z)e ^τz , τ ≤ 0} in the space H _σ ² (?₊), 0 < σ < + ∞, of functions analytic in the right-hand half-plane for which $$\parallel f\parallel : = \mathop {\sup }\limits_{ - \pi /2 < \varphi < \pi /2} \left\{ {\int_0^{ + \infty } {|f(re^{i\varphi } )|^2 } e^{ - 2r\sigma |\sin \varphi |} dr} \right\}^{1/2} < + \infty $$ .     

Bessel pairs and optimal Hardy and Hardy–Rellich inequalities

We give necessary and sufficient conditions on a pair of positive radial functions V and W on a ball B of radius R in R ⁿ , n ≥ 1, so that the following inequalities hold for all \({u \in C_{0}^{\infty}(B)}\) :

$\label{one} \int\limits_{B}V(x)|\nabla u |^{2}dx \geq \int\limits_{B} W(x)u^2dx,$

$\label{two} \int\limits_{B}V(x)|\Delta u |^{2}dx \geq\int\limits_{B} W(x)|\nabla u|^{2}dx+(n-1)\int\limits_{B}\left(\frac{V(x)}{|x|^2}-\frac{V_r(|x|)}{|x|}\right)|\nabla u|^2dx.$

This characterization makes a very useful connection between Hardy-type inequalities and the oscillatory behaviour of certain ordinary differential equations, and helps in the identification of a large number of such couples (V, W)—that we call Bessel pairs—as well as the best constants in the corresponding inequalities. This allows us to improve, extend, and unify many results—old and new—about Hardy and Hardy–Rellich type inequalities, such as those obtained by Caffarelli et al. (Compos Math 53:259–275, 1984), Brezis and Vázquez (Revista Mat. Univ. Complutense Madrid 10:443–469, 1997), Wang and Willem (J Funct Anal 203:550–568, 2003), Adimurthi et al. (Proc Am Math Soc 130:489–505, 2002), and many others.

     

Weakly k-hyponormal and polynomially hyponormal commuting operator pairs

We introduce the notion of weak k-hyponormality and polynomial hyponormality for commuting operator pairs on a Hilbert space and investigate their relationship with k-hyponormality and subnormality.We provide examples of 2-variable weighted shifts which are weakly 1-hyponormal but not hyponormal.By relating the weak k-hyponormality and k-hyponormality of a commuting operator pair to positivity of restriction of some linear functionals to corresponding cones of functions,we prove that there is an operator pair that is polynomially hyponormal but not 2-hyponormal,generalizing Curto and Putinar’s result(1991,1993)to the two-variable case.     

Lehmer pairs of zeros,the de Bruijn-Newman constant Λ, and the Riemann Hypothesis

We give here a rigorous formulation for a pair of consecutive simple positive zeros of the functionH ₀ (which is closely related to the Riemann -function) to be a Lehmer pair of zeros ofH ₀. With this formulation, we establish that each such pair of zeros gives a lower bound for the de Bruijn-Newman constant (where the Riemann Hypothesis is equivalent to the assertion that 0). We also numerically obtain the following new lower bound for :

Criteria for the divergence of pairs of Teichmüller geodesics

We study the asymptotic geometry of Teichmüller geodesic rays. We show that, when the transverse measures to the vertical foliations of the quadratic differentials determining two different rays are topologically equivalent, but are not absolutely continuous with respect to each other, the rays diverge in Teichmüller space.     

On Γ-convergence of pairs of dual functionals

The paper considers a slightly modified notion of the Γ-convergence of convex functionals in uniformly convex Banach spaces and establishes that under standard coercitivity and growth conditions the Γ-convergence of a sequence of functionals {Fj} to implies that the corresponding sequence of dual functionals converges in an analogous sense to the dual to functional .     

Whitham Equations, Bergman Kernel and Lax—Levermore Minimizer

We study multiphase solutions of the Whitham equations. The Whitham equations describe the zero dispersion limit of the Cauchy problem for the Korteweg—de Vries (KdV) equation. The zero dispersion solution of the KdV equation is determined by the Lax—Levermore minimization problem. The minimizer is a measurable function on the real line. When the support of the minimizer consists of a finite number of disjoint intervals to be determined, the minimization problem can be reduced to a scalar Riemann Hilbert (RH) problem. For each fixed x and t 0, the end-points of the contour are determined by the solution of the Whitham equations. The Lax—Levermore minimizer and the solution of the Whitham equations are described in terms of a kernel related to the Bergman kernel. At t = 0 the support of the minimizer consists of one interval for any value of x, while for t > 0, the number of intervals is larger than one in some regions of the (x,t) plane where the multiphase solutions of the Whitham equations develop. The increase of the number of intervals happens whenever the solution of the Whitham equations has a point of gradient catastrophe. For a class of smooth monotonically increasing initial data, we show that the support of the Lax—Levermore minimizer increases or decreases the number of its intervals by one near each point of gradient catastrophe. This result justifies the formation and extinction of the multiphase solutions of the Whitham equations. Furthermore we characterize a class of initial data for which all the points of gradient catastrophe occur only a finite number of times and therefore the support of the Lax—Levermore minimizer consists of a finite number of disjoint intervals for any x and t 0. This corresponds to give an upper bound to the genus of the solution of the Whitham equations. Similar results are obtained for the semi-classical limit of the defocusing nonlinear Schrödinger equation.     

The computation of Belyi pairs of 6-edged dessins d’enfants of genus 3 with automorphism groups of order 12 and 3

In this article, we study dessins d’enfants of genus 3 with six edges and only one vertex, having groups of automorphisms of order at least 3. We find all such dessins and explain the method of the enumeration. For each of them the Belyi pair is computed. Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 13, No. 6, pp. 137–148, 2007.     

Symmetries of Quantum Lax Equations for the Painlevé Equations

Based on the fact that the Painlevé equations can be written as Hamiltonian systems with affine Weyl group symmetries, a canonical quantization of the Painlevé equations preserving such symmetries has been studied recently. On the other hand, since the Painlevé equations can also be described as isomonodromic deformations of certain second-order linear differential equations, a quantization of such Lax formalism is also a natural problem. In this paper, we introduce a canonical quantization of Lax equations for the Painlevé equations and study their symmetries. We also show that our quantum Lax equations are derived from Virasoro conformal field theory.