A Note On Ill-posedness of the Cauchy Problem For Heisenberg Wave Maps

We introduce a notion of wave maps with a target in the sub- Riemannian Heisenberg group and study their relation with Riemannian wave maps with range in Lagrangian submanifolds. As an application we establish existence and eventually ill-posedness of the corresponding Cauchy problem

Stability theory of solitary waves in the presence of symmetry, II

AbstractConsider an abstract Hamiltonian system which is invariant under a group of operators. We continue to study the effect of the group invariance on the stability of solitary waves. Applications are given to bound states and traveling wave solutions of nonlinear wave equations

The role of sign indefinite invariants in shaping turbulent cascades

We highlight a non-canonical yet natural choice of variables for an efficient derivation of a kinetic equation for the energy density in non-isotropic systems, including internal gravity waves on a vertical plane, inertial and Rossby waves. The existence of a second quadratic invariant simplifies the kinetic equation and leads to extra conservation laws for resonant interactions. We analytically determine the scaling of the radial turbulent energy spectrum. Our findings suggest the existence of an inverse energy cascade of internal gravity waves, from small to large scales, in practically relevant scenarios