I. Exact solution of some nonlinear evolution equations. II. The similarity solution for the Korteweg-De Vries equation and the related Painleve Transcendent

In Part I, a method for finding solutions of certain diffusive dispersive nonlinear evolution equations is introduced. The method consists of a straightforward iteration procedure, applied to the equation as it stands (in most cases), which can be carried out to all terms, followed by a summation of the resulting infinite series, sometimes directly and other times in terms of traces of inverses of operators in an appropriate space.

We first illustrate our method with Burgers' and Thomas' equations, and show how it quickly leads to the Cole-Hopft transformation, which is known to linearize these equations.

We also apply this method to the Korteweg and de Vries, nonlinear (cubic) Schrödinger, Sine-Gordon, modified KdV and Boussinesq equations. In all these cases the multisoliton solutions are easily obtained and new expressions for some of them follow. More generally we show that the Marcenko integral equations, together with the inverse problem that originates them, follow naturally from our expressions.

Only solutions that are small in some sense (i.e., they tend to zero as the independent variable goes to ∞) are covered by our methods. However, by the study of the effect of writing the initial iterate u_1 = u_(1)(x,t) as a sum u_1 = ^∼/u_1 + ^≈/u_1 when we know the solution which results if u_1 = ^∼/u_1, we are led to expressions that describe the interaction of two arbitrary solutions, only one of which is small. This should not be confused with Backlund transformations and is more in the direction of performing the inverse scattering over an arbitrary “base” solution. Thus we are able to write expressions for the interaction of a cnoidal wave with a multisoliton in the case of the KdV equation; these expressions are somewhat different from the ones obtained by Wahlquist (1976). Similarly, we find multi-dark-pulse solutions and solutions describing the interaction of envelope-solitons with a uniform wave train in the case of the Schrodinger equation.

Other equations tractable by our method are presented. These include the following equations: Self-induced transparency, reduced Maxwell-Bloch, and a two-dimensional nonlinear Schrodinger. Higher order and matrix-valued equations with nonscalar dispersion functions are also presented.

In Part II, the second Painleve transcendent is treated in conjunction with the similarity solutions of the Korteweg-de Vries equat ion and the modified Korteweg-de Vries equation.

Conduction-electron localized-moment interaction in palladium-silicon base amorphous alloys containing transition metals

The electrical and magnetic properties of amorphous alloys obtained by rapid quenching from the liquid state have been studied. The composition of these alloys corresponds to the general formula M_xPd_80-xSi₂₀, in which M stands for a metal of the first transition series between chromium and nickel and x is its atomic concentration. The concentration ranges within which an amorphous structure could be obtained were: from 0 to 7 for Cr, Mn and Fe, from 0 to 11 for Co and from 0 to 15 for Ni. A well-defined minimum in the resistivity vs temperature curve was observed for all alloys except those containing nickel. The alloys for which a resistivity minimum was observed had a negative magnetoresistivity approximately proportional to the square of the magnetization and their susceptibility obeyed the Curie-Weiss law in a wide temperature range. For concentrated Fe and Co alloys the resistivity minimum was found to coexist with ferromagnetism. These observations lead to the conclusion that the present results are due to a s-d exchange interaction. The unusually high resistivity minimum temperature observed in the Cr alloys is interpreted as a result of a high Kondo temperature and a large s-d exchange integral. A low Fermi energy of the amorphous alloys (3.5 eV) is also responsible for the anomalies due to the s-d exchange interaction.