Riemann–Hilbert formulation for the KdV equation on a finite interval

The initial-boundary value problem for the KdV equation on a finite interval is analyzed in terms of a singular Riemann–Hilbert problem for a matrix-valued function in the complex k-plane which depends explicitly on the space–time variables. For an appropriate set of initial and boundary data, we derive the k-dependent “spectral functions” which guarantee the uniqueness of Riemann–Hilbert problem's solution. The latter determines a solution of the initial-boundary value problem for KdV equation, for which an integral representation is given. To cite this article: I. Hitzazis, D. Tsoubelis, C. R. Acad. Sci. Paris, Ser. I 347 (2009).     

Boundary Value Problems for the Elliptic Sine-Gordon Equation in a Semi-strip

We study boundary value problems posed in a semistrip for the elliptic sine-Gordon equation, which is the paradigm of an elliptic integrable PDE in two variables. We use the method introduced by one of the authors, which provides a substantial generalization of the inverse scattering transform and can be used for the analysis of boundary as opposed to initial-value problems. We first express the solution in terms of a 2×2 matrix Riemann–Hilbert problem whose “jump matrix” depends on both the Dirichlet and the Neumann boundary values. For a well posed problem one of these boundary values is an unknown function. This unknown function is characterised in terms of the so-called global relation, but in general this characterisation is nonlinear. We then concentrate on the case that the prescribed boundary conditions are zero along the unbounded sides of a semistrip and constant along the bounded side. This corresponds to a case of the so-called linearisable boundary conditions, however, a major difficulty for this problem is the existence of non-integrable singularities of the function q _y at the two corners of the semistrip; these singularities are generated by the discontinuities of the boundary condition at these corners. Motivated by the recent solution of the analogous problem for the modified Helmholtz equation, we introduce an appropriate regularisation which overcomes this difficulty. Furthermore, by mapping the basic Riemann–Hilbert problem to an equivalent modified Riemann–Hilbert problem, we show that the solution can be expressed in terms of a 2×2 matrix Riemann–Hilbert problem whose “jump matrix” depends explicitly on the width of the semistrip L, on the constant value d of the solution along the bounded side, and on the residues at the given poles of a certain spectral function denoted by h(λ). The determination of the function h remains open.     

The Camassa–Holm equation on the half-line

We study the initial-boundary-value problem for the Camassa–Holm equation on the half-line by associating to it a matrix Riemann–Hilbert problem in the complex k-plane; the jump matrix is determined in terms of the spectral functions corresponding to the initial and boundary values. We prove that if the boundary values $u(0,t)$ are ?0 for all t then the corresponding initial-boundary-value problem has a unique solution, which can be expressed in terms of the solution of the associated RH problem. In the case $u(0,t)<0$, the compatibility of the initial and boundary data is explicitly expressed in terms of an algebraic relation to be satisfied by the spectral functions. To cite this article: A. Boutet de Monvel, D. Shepelsky, C. R. Acad. Sci. Paris, Ser. I 341 (2005).     

A Fokas approach to the coupled modified nonlinear Schrödinger equation on the half‐line

In this work, we discuss the coupled modified nonlinear Schrödinger (CMNLS) equation, which describe the pulse propagation in the picosecond or femtosecond regime of the birefringent optical fibers. By use of the Fokas approach, the initial‐boundary value problem for the CMNLS equation related to a 3×3 matrix Lax pair on the half‐line is to be analyzed. Assuming that the solution {u(x,t),v(x,t)} of CMNLS equation exists, we will prove that it can be expressed in terms of the unique solution of a 3×3 matrix Riemann‐Hilbert problem formulated in the plane of the complex spectral parameter λ. Moreover, we also get that some spectral functions s(λ) and S(λ) are not independent of each other but meet a global relationship.     

On a new class of boundary value problems for the Sturm-Liouville operator

We consider the spectral problem generated by the Sturm-Liouville operator with complex-valued potential q(x) ∈ L ₂(0, π) and degenerate boundary conditions. We show that the set of potentials q(x) for which there exist associated functions of arbitrarily high order in the system of root functions is everywhere dense in L ₁(0, π).     

Algebraic braided model of the affine line and difference calculus on a topological spaceModèle algébrique tressé de la droite affine et calcul aux différences sur un espace topologique

In a previous Note [1], we suggested a quantum model of the unit interval [0,1], using convergent power series, parametrized by a variable q (a remarkable example is the quantum exponential, defined by Euler). In the present Note, we suggest a simpler model based on functions $\text{f=f(x):}\text{Z}\text{→k}$ (with an arbitrary commutative ring k) which are constant when x?+∞ or x??∞ and their “differentials” considered as functions x?f(x+1)?f(x) (difference calculus). Thanks to this new “differential calculus over the integers”, we can associate to any simplicial set or topological space X a braided differential graded algebra $\text{D}{}^1\text{(X)}$ which is similar in spirit to the algebra $\text{W}{}^1\text{(X)}$ introduced in [1]. We notice that the p-homotopy type of X can be read from the braiding of $\text{D}{}^1\text{(X)}$. In particular, if $\text{k=}\text{Z}$, we recover in a purely algebraic way the integral cohomology, Steenrod operations, homotopy groups from this braiding. To cite this article: M. Karoubi, C. R. Acad. Sci. Paris, Ser. I 335 (2002) 121–126.     

Modulated waves in a semiclassical continuum limit of an integrable NLS chain

A one‐dimensional integrable lattice system of ODEs for complex functions Q_n(τ) that exhibits dispersive phenomena in the phase is studied. We consider wave solutions of the local form Q_n(τ) ∼ qexp(i(kn + ωτ + c)), in which q, k, and ω modulate on long time and long space scales t = ετ and x = εn. Such solutions arise from initial data of the form Q_n(0) = q(nε) exp(iϕ(nε)/ε), the phase derivative ϕ′ 0 giving the local value of the phase difference k. Formal asymptotic analysis as ε → 0 yields a first‐order system of PDEs for q and ϕ′ as functions of x and t. A certain finite subchain of the discrete system is solvable by an inverse spectral transform. We propose formulae for the asymptotic spectral data and use them to study the limiting behavior of the solution in the case of initial data |Q_n| < 1, which yield hyperbolic PDEs in the formal limit. We show that the hyperbolic case is amenable to Lax‐Levermore theory. The associated maximization problem in the spectral domain is solved by means of a scalar Riemann‐Hilbert problem for a special class of data for all times before breaking of the formal PDEs. Under certain assumptions on asymptotic behaviors, the phase and amplitude modulation of the discrete systems is shown to be governed by the formal PDEs. Modulation equations after breaking time are not studied. Full details of the WKB theory and numerical results are left to a future exposition. © 2000 John Wiley & Sons, Inc.     

Non-conformal Repellers and the Continuity of Pressure for Matrix Cocycles

The pressure function P(A, s) plays a fundamental role in the calculation of the dimension of “typical” self-affine sets, where A = (A ₁, …,A _k ) is the family of linear mappings in the corresponding generating iterated function system. We prove that this function depends continuously on A. As a consequence, we show that the dimension of “typical” self-affine sets is a continuous function of the defining maps. This resolves a folklore open problem in the community of fractal geometry. Furthermore we extend the continuity result to more general sub-additive pressure functions generated by the norm of matrix products or generalized singular value functions for matrix cocycles, and obtain applications on the continuity of equilibrium measures and the Lyapunov spectrum of matrix cocycles.     

On the completeness of the system of root functions of the Sturm-Liouville operator with degenerate boundary conditions

We consider the spectral problem generated by the Sturm-Liouville equation with arbitrary complex-valued potential q(x) ∈ L ₁(0, π) and with degenerate boundary conditions. We obtain sufficient conditions for the completeness of the system of eigenfunctions and associated functions of this operator.     

An Initial-Boundary Value Problem for the Korteweg–de Vries Equation with Dominant Surface Tension

We consider the initial-boundary value problem (IBVP) for the Korteweg–de Vries equation with zero boundary conditions at x=0 and arbitrary smooth decreasing initial data. We prove that the solution of this IBVP can be found by solving two linear inverse scattering problems (SPs) on two different spectral planes. The first SP is associated with the KdV equation. The second SP is self-conjugate and its scattering function is found in terms of entries of the scattering matrix s(k) for the first SP. Knowing the scattering function, we solve the second inverse SP for finding the potential self-conjugate matrix. Consequently, the unknown object entering coefficients in the system of evolution equations for s(k,t) is found. Then, the time-dependent scattering matrix s(k,t) is expressed in terms of s(k)=s(k,0) and of solutions of the self-conjugate SP. Knowing s(k,t), we find the solution of the IBVP in terms of the solution of the Gelfand–Levitan–Marchenko equation in the first inverse SP.     

The RH boundary value problem of the k-monogenic functions

In this paper we study the Riemann and Hilbert problems of k-monogenic functions. By using Euler operator, we transform the boundary value problem of k-monogenic functions into the boundary value problems of monogenic functions. Then by the Almansi-type theorem of k-monogenic functions, we get the solutions of these problems.     

Finite generation of lifted p-adic homology with compact supports. Generalization of the weil conjectures to singular,non-complete algebraic varieties

In Section 1, if $\text{O}$ is a c.d.v.r. with quotient field of characteristic zero and residue class field k, if $\text{A}$ is an $\text{O}$-algebra and if $\text{A =}\text{A}$ ?_$\text{O}$k, then for algebraic families X over A that are polynomially properly embeddable over $\text{A}$, we define the lifted p-adic homology with compact supports$\text{H}{}_{\mathrm{h}}{}^{\mathrm{c}}\text{(X,}\text{A}$2 ?_zQ), which are functors with respect to proper maps. In Section 2, it is shown that, if X is an algebraic variety over k (i.e., if A = k), then the lifted p-adic homology of X with compact supports with coefficients in K is finite dimensional over K = quotient field of $\text{O}$. In Section 3, the results of Sections 1 and 2 are used to generalize both the statement and proof of the Weil “Lefschetz Theorem” Conjecture and the statement (but not the proof) of the Weil “Riemann Hypothesis” Conjecture, to non-complete, singular varieties over finite fields. In addition, the Weil zeta function of varieties over finite fields, is generalized by a device which we call the zeta matrices, W^h(X), 0 ≤ h ≤ 2 dim X, of an algebraic variety X, to varieties over even infinite fields of non-zero characteristic. These are used to give formulas for the zeta functions of each variety in an algebraic family, by means of the zeta matrices of an alebraic family. Sketches only are given. In Section 4, some of the material is duplicated, to define a q-adic homology with compact supports, q ≠ characteristic. The definition only makes sense for algebraic varieties; finite generation is proved. And the Weil “Lefschetz Theorem” Conjecture is established, even for singular, non-complete varieties, as well as a generalization of the Weil “Riemann Hypothesis” Conjecture. (However, zeta matrices do not make sense q-adically.)In Section 5, some special results are proved about p-adic homology with compact supports on affines. And the Weil “Riemann Hypothesis” conjecture is proved p-adically, p = characteristic, for projective, non-singular liftable varieties.     

Partial difference equations in m1⩾m2⩾⋯⩾mn⩾0 and their applications to combinatorics

Various discrete functions encountered in Combinatorics are solutions of Partial Difference Equations in the subset of Nⁿ given by m₁?m₂???m_n?0. Given a partial difference equation, it is described how to pass from the standard “easy” solution of an equation in Nⁿ to a solution of the same equation subject to certain “Dirichlet” or “Neumann” boundary conditions in the domain m₁?m₂???m_n?0 and related domains. Applications include a rather quick derivation of MacMahon's generating function for plane partitions, a generalization and q-analog of the Ballot problem, and a joint analog of the Ballot problem and Simon Newcomb's problem.     

Asymptotics of the spectrum of the Sturm-Liouville operator with regular boundary conditions

On the interval (0, π), we consider the spectral problem generated by the Sturm-Liouville operator with regular but not strongly regular boundary conditions. For an arbitrary potential q(x) ∈ L ₁ (0, π) [q(x) ∈ L ₂(0, π)], we establish exact asymptotic formulas for the eigenvalues of this problem.     

On the divergence of expansions in systems of root functions of the Sturm-Liouville operator with degenerate boundary conditions

We consider the spectral problem generated by the Sturm-Liouville operator with an arbitrary complex-valued potential q(x) ?? L ₁(0, ??) and with degenerate boundary conditions. We show that, under some additional condition, the system of root functions of that operator is not a basis in the space L ₂(0, ??).     

Nonlinear Fourier transforms for the sine-Gordon equation in the quarter plane

Using the Unified Transform, also known as the Fokas method, the solution of the sine-Gordon equation in the quarter plane can be expressed in terms of the solution of a matrix Riemann–Hilbert problem whose definition involves four spectral functions $a,b,A,B$. The functions $a(k)$ and $b(k)$ are defined via a nonlinear Fourier transform of the initial data, whereas $A(k)$ and $B(k)$ are defined via a nonlinear Fourier transform of the boundary values. In this paper, we provide an extensive study of these nonlinear Fourier transforms and the associated eigenfunctions under weak regularity and decay assumptions on the initial and boundary values. The results can be used to determine the long-time asymptotics of the sine-Gordon quarter-plane solution via nonlinear steepest descent techniques.     

Solution dependence on initial conditions in differential variational inequalities

In the first part of this paper, we establish several sensitivity results of the solution x(t, ξ) to the ordinary differential equation (ODE) initial-value problem (IVP) dx/dt = f(x), x(0) =  ξ as a function of the initial value ξ for a nondifferentiable f(x). Specifically, we show that for $\Xi_T \equiv \{\,x(t,\xi^0): 0 \leq t \leq T\,\}$ , (a) if f is “B-differentiable” on $\Xi_T$ , then so is the solution operator x(t;·) at ξ⁰; (b) if f is “semismooth” on $\Xi_T$ , then so is x(t;·) at ξ⁰; (c) if f has a “linear Newton approximation” on $\Xi_T$ , then so does x(t;·) at ξ⁰; moreover, the linear Newton approximation of the solution operator can be obtained from the solution of a “linear” differential inclusion. In the second part of the paper, we apply these ODE sensitivity results to a differential variational inequality (DVI) and discuss (a) the existence, uniqueness, and Lipschitz dependence of solutions to subclasses of the DVI subject to boundary conditions, via an implicit function theorem for semismooth equations, and (b) the convergence of a “nonsmooth shooting method” for numerically computing such boundary-value solutions.     

Uniqueness Theorems for Sturm–Liouville Operators with Boundary Conditions Polynomially Dependent on the Eigenparameter from Spectral Data

In this paper, we discuss the inverse problems for Sturm–Liouville operators with boundary conditions polynomially dependent on the spectral parameter. We establish some uniqueness theorems on the potential q(x) for the half inverse problem and the interior inverse problem from spectral data, respectively.     

Spectral integrals of operator-valued functions. II. From the study of stationary processes

This paper is a continuation of the study made in [38]. Using Douglas' operator range theorem and Crimmins' corollary we obtain several new results on the “square-integrability of operator-valued functions with respect to a nonnegative hermitian measure”. Using these facts we are able to extend in an important way theorems on the “spectral integral of an operator-valued function” which were obtained in [38], to wit, we are able to drop assumptions that functions are closed operator-valued. We apply these results to Wiener-Masani type infinite-dimensional stationary processes, representing a purely non-deterministic process as a “moving average” and obtaining a “factorization” of its spectral density. Next, anticipating global applications of our tools, we investigate the adjoint and generalized inverse of spectral integrals. Our definition of measurability for closed-operator-valued functions plays a key role here. Finally, we partially prove a conjecture (J. Multivariate Anal. (1974), 166–209) on simpler necessary and sufficient conditions on “when is a closed densely defined operator T from $\text{H}$^q to $\text{H}$^p a spectral integral T = fΦdE?”: Let q be finite and E be of countable multiplicity for $\text{H}$. Then (i) Tx ∈ $\text{S}$_x^p each x ∈ $\text{D}$_T (T is E-subordinate), and (ii) E(B)T ? TE(B) each B ∈ $\text{B}$ (T is E-commutative) implies L_x^pT ? TL_x^q each x ∈ $\text{H}$^q (T commutes with all the cyclic projections), and thus T = fΦdE.