Martin D. Kruskal

An Energy Principle for Hydromagnetic Stability Problems

The problem of the stability of static, highly conducting, fully ionized plasmas is investigated by means of an energy principle developed from one introduced by Lundquist. The derivation of the principle and the conditions under which it applies are given. The method is applied to find complete stability criteria for two types of equilibrium situations. The first concerns plasmas which are completely separated from the magnetic field by an interface. The second is the general axisymmetric system.

Korteweg‐de Vries Equation and Generalizations. II. Existence of Conservation Laws and Constants of Motion

With extensive use of the nonlinear transformations presented in Paper I of the series, a variety of conservation laws and constants of motion are derived for the Korteweg‐de Vries and related equations. A striking connection with the Sturm‐Liouville eigenvalue problem is exploited.

New similarity reductions of the Boussinesq equation

Some new similarity reductions of the Boussinesq equation, which arises in several physical applications including shallow water waves and also is of considerable mathematical interest because it is a soliton equation solvable by inverse scattering, are presented. These new similarity reductions, including some new reductions to the first, second, and fourth Painleve equations, cannot be obtained using the standard Lie group method for finding group‐invariant solutions of partial differential equations; they are determined using a new and direct method that involves no group theoretical techniques.

Equilibrium of a Magnetically Confined Plasma in a Toroid

A variety of properties are derived satisfied by any static equilibrium of a plasma governed by the well‐known magnetostatic equations. Some of these are local and quite trivial. Others involve integrals over surfaces of constant pressure, which are shown to be topologically toroidal under fairly general assumptions.A variational principle for such equilibria is derived. One of its consequences is to provide a characterization of equilibria by their values of certain invariants.Finally, conditions are obtained additional to the magnetostatic equations appropriate to the steady state of a plasma slowly diffusing across a magnetic field out of a topologically toroidal region.

Some Instabilities of a Completely Ionized Plasma

Two cases of equilibrium for a highly conducting plasma are investigated for their stability. In the first case, a plasma is supported against gravity by the pressure of a horizontal magnetic field. This equilibrium is found unstable, in close correspondence to the classical case of a heavy fluid supported by a light one. The second case refers to the so-called pinch effect. Here a plasma is kept within a cylinder by the pressure of a toroidal magnetic field which in turn is caused by an electric current within the plasma. This equilibrium is found unstable against lateral distortions.

On the Stability of Plasma in Static Equilibrium

Criteria useful for the investigation of the stability of a system of charged particles are derived from the Boltzmann equation in the small m/e limit. These criteria are obtained from the examination of the variation of the energy due to a perturbation, when subject to the general constraint that all regular, time‐independent constants of the motion (including the energy) have their equilibrium values.The first‐order variation of the energy vanishes trivially, and the second‐order variation yields a quadratic form in the displacement variable ξ (which may be introduced because of the well‐known properties of this limit). The positive‐definiteness of this form is a sufficient condition for stability.Several theorems are stated comparing stability under the present theory with that under conventional hydromagnetic fluid theories where heat flow along magnetic lines of force is neglected. Generalizations can be made to systems where the effect of collisions is included.

Korteweg‐deVries Equation and Generalizations. V. Uniqueness and Nonexistence of Polynomial Conservation Laws

The conservation laws derived in an earlier paper for the Korteweg‐deVries equation are proved to be the only ones of polynomial form. An algebraic operator formalism is developed to obtain explicit formulas for them.

Painlevé test for the self-dual Yang-Mills equation

Abstract It is shown that the SU(2) self-dual Yang-Mills equation passes the Painleve test for complete integrability.

Solitary Wave Collisions

The interactions of nonperiodic solitary waves are numerically investigated for the nonlinear Klein–Gordon equation. It is found that the collisions are generally inelastic. Special solutions to the sine-Gordon equation are discussed.

Hydromagnetic Instability in a Stellarator

Stability diagrams are calculated for hydromagnetic kink instabilities in a zero‐pressure plasma confined in a stellarator geometry with arbitrary mixtures of l = 2 and l = 3 multipolar helical fields and with various radial distributions of the current parallel to the confining magnetic field. The introduction of small pressure gradients increases the size of the unstable regions. Modes with small azimuthal wavenumber grow more rapidly than those with large m. Experimental data, obtained with the model C Stellarator, show that the plasma is macroscopically stable except for certain intervals of rotational transform. These intervals agree qualitatively with those in which the theory predicts the plasma should be unstable against the kink instability.