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Martin David Kruskal was one of the most versatile theoretical physicists of his generation and is distinguished for his enduring work in several different areas, most notably plasma physics, a memorable detour into relativity and his pioneering work in nonlinear waves. In the latter, together with Norman Zabusky, he invented the concept of the soliton and, with others, developed its application to classes of partial differential equations of physical significance.

Inverse length scales for the 2d and 3d thermal convection equations on periodic boundary conditions are estimated directly from the PDEs using a ladder theorem. To within logarithmic dependence on the Rayleigh number RT, the 2d estimate approximately agrees with the inverse length scale estimate found by Manley, Foias and Temam(1987) which was based on the attractor dimension. In keeping with numerical experiments, it shows that RT1/2*RT1/2 grid points should be roughly enough to resolve the smallest features in the flow. Similar methods applied to the 3d case produce a curious result where an inverse length scale for the flow can be estimated rigorously for weak solutions. This estimate is proportional to RT2 which has an associated length scale which is unfortunately much too small to be of any practical use. The same methods also apply to double diffusive convection.

Two possible diagnostics of stretching and folding (S&F) in fluid flows are discussed, based on the dynamics of the gradient of potential vorticity ($q = \bom\cdot\nabla\theta$) associated with solutions of the three-dimensional Euler and Navier-Stokes equations. The vector $\bdB = \nabla q \times \nabla\theta$ satisfies the same type of stretching and folding equation as that for the vorticity field $\bom $ in the incompressible Euler equations (Gibbon & Holm, 2010). The quantity $\theta$ may be chosen as the potential temperature for the stratified, rotating Euler/Navier-Stokes equations, or it may play the role of a seeded passive scalar for the Euler equations alone. The first discussion of these S&F-flow diagnostics concerns a numerical test for Euler codes and also includes a connection with the two-dimensional surface quasi-geostrophic equations. The second S&F-flow diagnostic concerns the evolution of the Lamb vector $\bsD = \bom\times\bu$, which is the nonlinearity for Euler's equations apart from the pressure. The curl of the Lamb vector ($\boldsymbol{\varpi} := \bsD$) turns out to possess similar stretching and folding properties to that of the $\bdB$-vector.

More than 150 years after their invention by Hamilton, quaternions are now widely used in the aerospace and computer animation industries to track the paths of moving objects undergoing three-axis rotations. It is shown here that they provide a natural way of selecting an appropriate ortho-normal frame -- designated the quaternion-frame -- for a particle in a Lagrangian flow, and of obtaining the equations for its dynamics. How these ideas can be applied to the three-dimensional Euler fluid equations is then considered. This work has a bearing on the issue of whether the Euler equations develop a singularity in a finite time. Some of the literature on this topic is reviewed, which includes both the Beale-Kato-Majda theorem and associated work on the direction of vorticity by both Constantin, Fefferman & Majda and Deng, Hou and Yu. It is then shown how the quaternion formulation provides a further direction of vorticity result using the Hessian of the pressure., 22 pages. This review is based on an invited lecture given at the meeting Mathematical Hydrodynamics held at the Steklov Institute Moscow, in June 2006

We present analytical methods whereby weak and strong turbulence are predicted in the D dimensional complex Ginzburg Landau (CGL) equation \(A_{t}=RA+(1+i\nu)\Delta A-(1+i\mu)A\vert A\vert^{2q}\) on a periodic domain [0,1]. Strong (hard) turbulence is characterised by large fluctuations away from space & time averages while no such fluctuations occur in weak (soft) turbulence. In the Δ-ν plane, there are different areas where weak & strong behaviour can occur. In the strong case (Δ, ν → ±∞, ‡∞), the corresponding areas go out to the inviscid limit where the CGL equation becomes the NLS equation in which a finite time singularity occurs when qD ≥ 2. A new infinite set of differential inequalities for the “lattice” of functionals \(F_{n,m}=\int\left[{\vert\nabla^{n-1}A\vert}^{2m}+\alpha_{n,m}\vert A\vert^{2m\left[q(n-1)+1\right]}\right]d\underline{x}\) enables us to construct large time upper bounds on the F n,m . The occurrence of strong spiky turbulence is predicted for qD = 2 by showing that exponents of R in the upper bounds of F n,m & ‖A‖∞ in the strong regions are dependent on the quantity |ν| which gets large in the inviscid limit. The critical value qD = 2 plays an important role: when qD > 2 the CGL equation has some similarities with the 3D Navier equations. A comparison is made between the two & the possibility of having a 1D system which mimics some limited features of the Navier Stokes equations is discussed.

A method is described which enables multiple three-dimensional soliton-like solutions to be found for non-linear Klein-Gordon equations of the type Square Operator phi =F( phi ). The method is extendable to coupled equations Square Operator phi =F( phi , psi ); Square Operator psi =G( phi , psi ).

We consider solutions of the equation\(\psi \), where\(\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{L} = \alpha ^{ij} \partial _i \partial _j (i,j = 0,...,n)\) and where α is any constant symmetric matrix. The solutions behave as soliton-shaped wave fronts inn dimensions. These interact but keep their identity as in one dimension. We give solutions for the interaction of up to four of these waves for any matrix α and suggest a solution forN of them. There is only one essential difference between these solutions and those of the one-dimensional system;i.e. conditions are found which restrict the motion of then-dimensional waves and it is these conditions we find the most interesting. If we choose α=diag (−1, 1, 1), then it is found that any three interacting waves intersect and form a triangle. For this α we find that the area of this triangle must remain constant in time and hence the wave fronts are forced to move in a specific fixed configuration. We also briefly consider a type of two-dimensional Korteweg-de Vries equation which, although artificially constructed from the one-dimensional case, has similar solutions to the 2-dimensional sine-Gordon equation.