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A discussion of the theory and applications of classical solitons is presented with a brief treatment of quantum mechanical effects which occur in particle physics and quantum field theory. The… (More)

1. The equations of motion 2. Dimensionless parameters and stability 3. Turbulence 4. Degrees of freedom, dynamical systems and attractors 5. On the existence, uniqueness and regularity of solutions… (More)

- P. J. Caudrey, R. K. Dodd, John D. Gibbon
- Proceedings of the Royal Society of London. A…
- 1976

We have found new hierarchies of Korteweg–de Vries and Boussinesq equations which have multiple soliton solutions. In contrast to the standard hierarchy of K. de V. equations found by Lax, these… (More)

- John D. Gibbon
- 2008

The three-dimensional Euler equations have stood for a quarter of a millenium as a challenge to mathematicians and physicists. While much has been discovered, the nature of solutions is still largely… (More)

We study numerically a class of stretched solutions of the three-dimensional Euler and Navier–Stokes equations identified by Gibbon, Fokas, and Doering (1999). Pseudo-spectral computations of a Euler… (More)

- John D. Gibbon
- 2012

Two unusual time-integral conditional regularity results are presented for the three-dimensional Navier-Stokes equations. The ideas are based on L2m-norms of the vorticity, denoted by Ωm(t), and… (More)

In this work we apply some recent developments in the analysis of the Navier–Stokes equations (Doering & Foias 2002) to mixing and the advection–diffusion equation. Mixing phenomena are ubiquitous… (More)

Abstract A well known limitation with stretched vortex solutions of the 3D Navier–Stokes (and Euler) equations, such as those of Burgers type, is that they possess uni-directional vorticity which is… (More)

We have undertaken a study of the complex Lorenz equations
x = −σx + σy
.
y = (r − z)x − ay
.
z = −bz + 12(x∗y + xy∗)
. where x and y are complex and z is real. The complex parameters r and a… (More)

The complex Ginzburg-Landau equation in one spatial dimension with periodic boundary conditions is studied from the viewpoint of effective low-dimensional behaviour by three distinct methods. Linear… (More)