A systematic method is developed which allows one to identify certain important classes of evolution equations which can be solved by the method of inverse scattering. The form of each evolution equation is characterized by the dispersion relation of its associated linearized version and an integro‐differential operator. A comprehensive presentation of the inverse scattering method is given and general features of the solution are discussed. The relationship of the scattering theory and Backlund transformations is brought out. In view of the role of the dispersion relation, the comparatively simple asymptotic states, and the similarity of the method itself to Fourier transforms, this theory can be considered a natural extension of Fourier analysis to nonlinear problems.

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A method of solution for the ’’derivative nonlinear Schrödinger equation’’ iqt=−qxx±i (q*q2)x is presented. The appropriate inverse scattering problem is solved, and the one-soliton solution is obtained, as well as the infinity of conservation laws. Also, we note that this equation can also possess ’’algebraic solitons.’’

Abstract. We derive a weak turbulence formalism for incompressible magnetohydrodynamics. Three-wave interactions lead to a system of kinetic equations for the spectral densities of energy and helicity. The kinetic equations conserve energy in all wavevector planes normal to the applied magnetic field B 0ê . Numerically and analytically, we find energy spectra E ± ∼ k n± ⊥ , such that n + + n − = −4, where E ± are the spectra of the Elsässer variables z ± = v ± b in the two-dimensional case (k = 0). The constants of the spectra are computed exactly and found to depend on the amount of correlation between the velocity and the magnetic field. Comparison with several numerical simulations and models is also made.

A method for solving certain nonlinear ordinary and partial differential equations is developed. The central idea is to study monodromy preserving deformations of linear ordinary differential equations with regular and irregular singular points. The connections with isospectral deformations and with classical and recent work on monodromy preserving deformations are discussed. Specific new results include the reduction of the general initial value problem for the Painleve equations of the second type and a special case of the third type to a system of linear singular integral equations. Several classes of solutions are discussed, and in particular the general expression for rational solutions for the second Painleve equation family is shown to be -ln(A + /A__\ where Δ + and Δ_ are determinants. We also demonstrate that each of these equations is an exactly integrable Hamiltonian system.* The basic ideas presented here are applicable to a broad class of ordinary and partial differential equations additional results will be presented in a sequence of future papers.

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The main purpose of this work is to show how a continuous finite bandwidth of modes can be readily incorporated into the description of post-critical Rayleigh-Bénard convection by the use of slowly varying (in space and time) amplitudes. Previous attempts have used a multimodal discrete analysis. We show that in addition to obtaining results consistent with the discrete mode approach, there is a larger class of stable and realizable solutions. The main feature of these solutions is that the amplitude and wave-number of the motion is that of the most unstable mode almost everywhere, but, depending on external and initial conditions, the roll couplets in different parts of space may be 180° out of phase. The resulting discontinuities are smoothed by hyperbolic tangent functions. In addition, it is clear that the mechanism for propagating spatial nonuniformities is diffusive in character.

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