In this paper, we classify the four-dimensional gradient shrinking solitons under certain curvature conditions satisfied by all solitons arising from finite-time singularities of Ricci flow on compact four-manifolds with positive isotropic curvature. As a corollary, we generalize a result of Perelman on three-dimensional gradient shrinking solitons to dimension four.

We address the properties of surface solitons supported by optical lattices imprinted in photorefractive media with asymmetrical diffusion nonlinearity. Such solitons exist only in finite gaps of the lattice spectrum. In contrast to latticeless geometries, where surface waves exist only when nonlinearity deflects light toward the material surface, the surface lattice solitons exist in settings ...

We study spatiotemporal dynamics of soliton-induced twooctave- broad supercontinuum generated by fs pulses in an array of coupled nonlinear waveguides. We show that after fission of the input pulse into several fundamental solitons, red and blue-shifted nonsolitonic radiation, as well as solitons with lower intensity, spread away in transverse direction, while the most intense spikes self-trap ...

This is a review paper of basic knowledge and recent advances in the area of spatial solitons in Kerr and Kerr-like media. We consider spatial bright and dark solitons, sol-itons in waveguide geometries, optical bullets, vortex solitons and, brie¯y, dissipative cases. In our treatment, we use a Hamiltonian approach when considering stability issues.

A model for a non-Kerr cylindrical nematic fiber is presented. We use the multiple scales method to show the possibility of constructing different kinds of wave packets of transverse magnetic modes propagating through the fiber. This procedure allows us to generate different hierarchies of nonlinear partial differential equations which describe the propagation of optical pulses along the fiber....

Solitons are proposed as the agents of plastic and viscoelastic deformation in aligned polyethylene. Interactions between straight, parallel molecules are mapped rigorously onto the Frenkel-Kontorova model. It is shown that these molecular interactions distribute an applied load between molecules, with a characteristic transfer length equal to the soliton width. Load transfer leads to the intro...

We present a comprehensive study of interactions (collisions) between two-dimensional composite vector solitons carrying topological charges in isotropic saturable nonlinear media. We numerically study interactions between such composite solitons for different regimes of collision angle and report numerous effects which are caused solely by the "spin" (topological charge) carried by the second ...

We study the instability of algebraic solitons for integrable nonlinear equations in one spatial dimension that include modified KdV, focusing NLS, derivative NLS, and massive Thirring equations. We develop the analysis of the Evans function that defines eigenvalues in the corresponding Lax operators with algebraically decaying potentials. The standard Evans function generically has singulariti...

An explicit two-soliton solution for the derivative nonlinear Schrödinger equation with nonvanishing boundary conditions is derived, demonstrating details of interactions between two bright solitons, two dark solitons, as well as one bright soliton and one dark soliton. Shifts of soliton positions due to collisions are analytically obtained, which are irrespective of the bright or dark characte...

In this paper we classify the four dimensional gradient shrinking solitons under certain curvature conditions satisfied by all solitons arising from finite time singularities of Ricci flow on compact four manifolds with positive isotropic curvature. As a corollary we generalize a result of Perelman on three dimensional gradient shrinking solitons to dimension four.