Caroline Verhoeven

On a generalization of the iterative soft-thresholding algorithm for the case of non-separable penalty

An explicit algorithm for the minimization of an l1-penalized least-squares functional, with non-separable l1 term, is proposed. Each step in the iterative algorithm requires four matrix vector multiplications and a single simple projection on a convex set (or equivalently thresholding). Convergence is proven and a 1/N convergence rate is derived for the functional. In the special case where the matrix in the l1 term is the identity (or orthogonal), the algorithm reduces to the traditional iterative soft-thresholding algorithm. In the special case where the matrix in the quadratic term is the identity (or orthogonal), the algorithm reduces to a gradient projection algorithm for the dual problem. By replacing the projection with a simple proximity operator, other convex non-separable penalties than those based on an l1-norm can be handled as well.

Explicit integration of the Hénon-Heiles Hamiltonians

Abstract We consider the cubic and quartic Hénon-Heiles Hamiltonians with additional inverse square terms, which pass the Painlevé test for only seven sets of coefficients. For all the not yet integrated cases we prove the singlevaluedness of the general solution. The seven Hamiltonians enjoy two properties: meromorphy of the general solution, which is hyperelliptic with genus two and completeness in the Painlevé sense (impossibility to add any term to the Hamiltonian without destroying the Painlevé property).

Nonlinear superposition formula for the Kaup-Kupershmidt partial differential equation

Abstract The fifth order Kaup–Kupershmidt (KK) equation is one of the solitonic equations related to the integrable cases of the Henon–Heiles system. As opposed to the Sawada–Kotera equation which is its dual equation, the construction of the N -soliton solutions of KK is not an easy task in using a perturbation scheme or the Hirota bilinear formalism. From the Backlund transformation obtained by singularity analysis considerations, we here establish the permutability theorem for the KK equation. This allows us to explain the “anomalous” structure of the N -soliton solution, empirically obtained by Parker [Physica D 137 (2000) 25–33, 34–48].

Integration of a generalized Hénon–Heiles Hamiltonian

The generalized Henon–Heiles Hamiltonian H=1/2(PX2+PY2+c1X2+c2Y2)+aXY2−bX3/3 with an additional nonpolynomial term μY−2 is known to be Liouville integrable for three sets of values of (b/a,c1,c2). It has been previously integrated by genus two theta functions only in one of these cases. Defining the separating variables of the Hamilton–Jacobi equations, we succeed here, in the two other cases, to integrate the equations of motion with hyperelliptic functions.

Soliton solutions of two bidirectional sixth-order partial differential equations belonging to the KP hierarchy

In this letter, we analyse two bidirectional sixth-order partial differential equations, which are reductions in (1 + 1) dimensions of equations belonging to the KP hierarchy. They have fourth-order and fifth-order Lax pairs, respectively. We derive their Backlund transformations and, from the nonlinear superposition formula, we can build their soliton solutions like a Grammian. The interesting dynamics of these solitons is that they may describe not only the overtaking collision but also the head-on collision of solitary waves of different type and shape.

Iterative algorithms for total variation-like reconstructions in seismic tomography

A qualitative comparison of total variation like penalties (total variation, Huber variant of total variation, total generalized variation, . . .) is made in the context of global seismic tomography. Both penalized and constrained formulations of seismic recovery problems are treated. A number of simple iterative recovery algorithms applicable to these problems are described. The convergence speed of these algorithms is compared numerically in this setting. For the constrained formulation a new algorithm is proposed and its convergence is proven.

An integrable hierarchy with a perturbed Hénon-Heiles system

We consider an integrable scalar partial differential equation (PDE) that is second order in time. By rewriting it as a system and applying the Wahlquist–Estabrook prolongation algebra method, we obtain the zero curvature representation of the equation, which leads to a Lax representation in terms of an energy-dependent Schrodinger spectral problem of the type studied by Antonowicz and Fordy. The solutions of this PDE system, and of its associated hierarchy of commuting flows, display weak Painleve behaviour, i.e. they have algebraic branching. By considering the travelling wave solutions of the next flow in the hierarchy, we find an integrable perturbation of the case (ii) Henon–Heiles system which has the weak Painleve property. We perform separation of variables for this generalized Henon–Heiles system and describe the corresponding solutions of the PDE.

Completeness of the cubic and quartic Hénon-Heiles Hamiltonians

The quartic Henon-Heiles Hamiltonian passes the Painleve test for only four sets of values of the constants. Only one of these, identical to the traveling-wave reduction of the Manakov system, has been explicitly integrated (Wojciechowski, 1985), while the other three have not yet been integrated in the general case (α, β, γ) ≠ (0, 0, 0). We integrate them by building a birational transformation to two fourth-order first-degree equations in the Cosgrove classiffication of polynomial equations that have the Painleve property. This transformation involves the stationary reduction of various partial differential equations. The result is the same as for the three cubic Henon-Heiles Hamiltonians, namely, a general solution that is meromorphic and hyperelliptic with genus two in all four quartic cases. As a consequence, no additional autonomous term can be added to either the cubic or the quartic Hamiltonians without destroying the Painleve integrability (the completeness property).

CKP and BKP equations related to the generalized quartic Hénon-Heiles Hamiltonian

In their classification of soliton equations from a group theoretical standpoint according to the representation of infinite Lie algebras, Jimbo and Miwa listed bilinear equations of low degree for the KP and the modified KP hierarchies. In this list, we consider the (1+1)-dimensional reductions of three particular equations of special interest for establishing some new links with the generalized Hénon–Heiles Hamiltonian, possibly useful for integrating the latter with functions having the Painlevé property. Two of those partial differential equations have N-soliton solutions that, as for the Kaup–Kupershmidt equation, can be written as the logarithmic derivative of a Grammian. Moreover, they can describe head-on collisions of solitary waves of different type and shape.

General solution for Hamiltonians with extended cubic and quartic potentials

In terms of hyperelliptic functions, we integrate a two-particle Hamiltonian with quartic potential and additional linear and nonpolynomial terms in the Liouville integrable cases 1 : 6 : 1 and 1 : 6 : 8.