Franklin Lambert

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

Abstract A direct and unifying scheme for the disclosure of bilinear Backlund 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.

Construction of Bäcklund Transformations with Binary Bell Polynomials

A method for constructing Backlund transformations (BTs) for sech-squared soliton systems is presented as an alternative to the usual procedure based on the bilinear exchange formalism. It is shown to work without the need of a bilinear representation in the case of a nonlocal Boussinesq equation. The obtained bilinear BT produces a modified nonlocal Boussinesq equation.

On the Hirota Representation of Soliton Equations with One Tau-Function

Alternative Hirota representations in terms of a single tau-function are derived for a variety of soliton equations, including the sine-Gordon and the Tzitzeica equations. The relevance of these representations with respect to known bilinear representations of integrable hierarchies is briefly discussed. The essentials of the derivation method are presented.

On modified NLS, Kaup and NLBq equations: differential transformations and bilinearization

New transformations between the nonlinear Schrodinger, Kaup and non-local Boussinesq equations as well as their modified counterparts are found and analysed. The bilinear representations of these equations, including an alternative bilinear form of the Chen - Lee - Liu equation, are obtained by a direct method based on the Bells exponential polynomials. Explicit Wronskian solutions to these equations are also presented.

CANONICAL BILINEAR SYSTEMS AND SOLITON RESONANCES

A direct method, based on the use of Hirota’s binary operators, is developed for the search of families of soliton equations. A crucial step is the explicit construction of canonical bilinear equations from which recursion-operators and consequently canonical Backlund-transformations and (or) Lax-pairs can be deduced. This construction is illustrated in the case of the KdV-, Sawada-Kotera (SK)- and mKdV-hierarchies.

Scattering at low energies and the inverse problem at fixed angular momentum

Abstract We consider the inverse formalism at fixed angular momentum and examine how to associate a meaningful regular potential with a given number of scattering parameters in the effective range expansion of an S-wave phase shift. Under the assumption that there are no bound states, we expand k cotg δ ( k ) as a continued fraction in k 2 and show that the necessary condition δ (∞)− Δ (0) = 0 is sufficient in order that a given [ N / M ] Pade approximant of k cotg δ ( k ), with M ⩽ N − 1, be exactly reproducible by a regular Bargmann potential V N B ( r ). We derive explicit conditions on the coefficients of the 3 lower [ N / M ] Pade approximants (and thus on the first four scattering parameters) for the existence of unique V 1,2 B potentials which reproduce exactly such extrapolations. These potentials are found to be stable against the noise produced by experimental errors on the scattering parameters.

Padé approximation of two-point functions and the calculation of resonances

Abstract The application of the method of Pade approximants to two-point functions of spin 0 and spin 1 2 is discussed in connection with the calculation of resonances in renormalizable Lagrangian field theory. The resonances which are considered belong to the Peierls class: they are excited states of a particle, with the same quantum numbers as the particle except parity, corresponding to complex poles of the propagator on an unphysical Riemann sheet of the energy plane. The diagonal Pade approximants are found to be particularly suited for approximating the mass function in the modified propagator, with regard to consistency wit the Kallen-Lehmann representation. Contrary to perturbative approximation, they open a way for avoiding ghost difficulties, yielding a finite wave renormalization constant. The analytic continuation of the propagator in the Pade approximation scheme is derived from that of the perturbation theory terms. A resonance equation is obtained on the real axis, the roots of which provide information on the masses of the possible resonances. The [1,1] Pade approximant is considered in the particular cases of the V-propagator in the Lee model and the nucleon propagator in the Yukawa model.

An Elementary Approach to Hierarchies of Soliton Equations

In this paper we present a systematic and elementary construction of the Hirota equations (in an equivalent formulation) that make up the bilinear KP and modified KP hierarchies. Our construction leads to a natural gradation of the Hirota equations of a given weight in each bilinear hierarchy, providing useful insight into their mutual relationship.

On Soliton Instabilities In 1+1 Dimensional Integrable Systems

One space dimensional soliton resonances are investigated for a family of integrable equations which are related by translation to the Sawada-Kotera hierarchy. These resonances are found to induce a structural instability responsible for the decay of solitons with amplitude belonging to a definite interval.

Padé approximation of the nucleon propagator and the calculation of the N∗(I=12, JP=12±) resonances

Abstract The method of Pade Approximants (P.A.) is applied to the nucleon propagator in the Yukawa-model, for calculating excited states of the nucleon with the same quantum numbers as the nucleon, except parity. The [1, 1] P.A. of the propagator is constructed from perturbative contributions to the mass-operator up to the fourth-order in the renormalized π N coupling constant. The elastic contributions are calculated analytically and expressed in terms of di- and tri-logarithms. The inelastic contributions are evaluated in the approximation in which the three-body imaginary parts are taken at zero pion-mass ( μ =0). A resonance equation is numerically solved on the real axis yielding resonance candidates for the S 11 - and P 11 -waves.