Elliptic equations with low regularity boundary data via the unified method

We use novel integral representations developed by the second author to prove certain rigorous results concerning elliptic boundary value problems in convex polygons. Central to this approach is the so-called global relation, which is a non-local equation in the Fourier space that relates the known boundary data to the unknown boundary values. Assuming that the global relation is satisfied in the weakest possible sense, i.e. in a distributional sense, we prove there exist solutions to Dirichlet, Neumann and Robin boundary value problems with distributional boundary data. We show that the analysis of the global relation characterises in a straightforward manner the possible existence of both integrable and non-integrable corner singularities.     

Enantioselective synthesis of (-)-dihydrocodeinone: a short formal synthesis of (-)-morphine

[reaction: see text] The radical cyclization approach to the morphine alkaloids has been applied in an asymmetric synthesis of (-)-dihydrocodeinone. A chiral cyclohexenol (R-32), from the CBS reduction of the enone, is the source of chirality. The first key step, tandem closure in which stereochemistry is controlled by geometric constraints, (-)-15b --> (+)-16, was followed by an unprecedented reductive hydroamination, completing the synthesis of (-)-dihydroisocodeine ((-)-17) in 13 steps from commercially available materials.     

A new transform method for evolution partial differential equations

We introduce a new transform method for solving initial-boundary-valueproblems for linear evolution partial differential equationswith spatial derivatives of arbitrary order. This method isillustrated by solving several such problems on the half-line{t > 0, 0 < x < }, and on the quarter-plane {t >0, 0 < x_j < , j = 1, 2}. For equations in one space dimensionthis method constructs q(x, t) as an integral in the complexk-plane involving an x-transform of the initial condition anda t-transform of the boundary conditions. For equations in twospace dimensions it constructs q(x₁, x₂, t) as an integral inthe complex (k₁, k₂)-planes involving an (x₁, x₂)-transformof the initial condition, an (x₂, t)-transform of the boundaryconditions at x₁ = 0, and an (x₁, t)-transform of the boundaryconditions at x₂ = 0. This method is simple to implement andyet it yields integral representations which are particularlyconvenient for computing the long time asymptotics of the solution.     

A novel tunable aromatic bromination method using alkyl bromides and sodium hydride in DMSO

Aromatic bromination on various aromatic systems with different substitutions was performed in the presence of alkyl bromide and sodium hydride in DMSO. Mono-bromination on a wide range of substrates was achieved by selecting proper alkyl bromides and controlling its amount. Further bromination could happen with more active alkyl bromides and additional amount of bromides and sodium hydride. The yields ranged from moderate to excellent. In addition, reaction mechanism was postulated to explain our observations.     

A semi-analytical numerical method for solving evolution and elliptic partial differential equations

A new method for analyzing initial–boundary value problems for linear and integrable nonlinear partial differential equations (PDEs) has been introduced by one of the authors.     

Analysis of the Global Relation for the Nonlinear Schrödinger Equation on the Half-line

It has been shown recently that the unique, global solution of the Dirichlet problem of the nonlinear Schrödinger equation on the half-line can be expressed through the solution of a 2×2 matrix Riemann–Hilbert problem. This problem is specified by the spectral functions {a(k),b(k)} which are defined in terms of the initial condition q(x,0)=q ₀(x), and by the spectral functions {A(k),B(k)} which are defined in terms of the specified boundary condition q(0,t)=g ₀(t) and the unknown boundary value q _x (0,t)=g ₁(t). Furthermore, it has been shown that given q ₀ and g ₀, the function g ₁ can be characterized through the solution of a certain 'global relation' coupling q ₀, g ₀, g ₁, and (t,k), where satisfies the t-part ofthe associated Lax pair evaluated at x=0. We show here that, by using a Gelfand–Levitan–Marchenko triangular representation of , the global relation can be explicitly solved for g ₁.     

Integrability of linear and nonlinear evolution equations and the associated nonlinear fourier transforms

The inverse spectral method is a nonlinear Fourier transform method for solving certain equations. Here, we emphasize that such transforms should be considered in their own right. We also elucidate further the connection between the Fourier transform and inverse spectral methods by establishing that linear equations can also be solved through the inverse spectral method.     

The Korteweg-de Vries equation and beyond

We review a new method for linearizing the initial-boundary value problem of the KdV on the semi-infinite line for decaying initial and boundary data. We also present a novel class of physically important integrable equations. These equations, which include generalizations of the KdV, of the modified KdV, of the nonlinear Schrödinger and of theN-wave interactions, are as generic as their celebrated counterparts and, furthermore it appears that they describe certain physical situations more accurately.     

The dressing method and nonlocal Riemann-Hilbert problems

Summary We consider equations in 2+1 solvable in terms of a nonlocal Riemann-Hilbert problem and show that for such an equation there exists a unified dressing method which yields: (i) a Lax pair suitable for obtaining solutions that are perturbations of an arbitrary exact solution of the given equation; (ii) certain integrable generalizations of the given equation. Using this generalized dressing method large classes of solutions of these equations, including dromions and line dromions, can be obtained. The method is illustrated by using theN-wave interactions, the Davey-Stewartson I, and the Kadomtsev-Petviashvili I equations. We also show that a careful application of the usual dressing method yields a certain generalization of theN-wave interactions.