We derive a new four-dimensional partial differential equation with the isospectral Lax representation by shrinking symmetry algebra of reduced quasi-classical self-dual Yang–Mills and applying technique twisted extensions to obtained Lie algebra. Then we find recursion operator for symmetries construct Bäcklund transformation between this Martínez Alonso–Shabat equation. Finally, integrable hi...

We consider classical solutions of the inviscid Surface Quasi-geostrophic equation that are a small perturbation $\epsilon$ from radial stationary solution $\theta=|x|$. use modified energy method to prove existence time $\frac{1}{\epsilon}$ scale $\frac{1}{\epsilon^4}$. Moreover, by perturbing in suitable direction we construct global smooth solutions, via bifurcation, rotate uniformly and spa...

We show that the eigenvalues of the first order partial differential equation derived by quasi-classical approximation of the Schrödinger equation can be computed from the trace of a classical operator. The derived trace formula is different from the Gutzwiller trace formula. The Fredholm determinant of the new operator is an entire function of the complex energy plane in contrast to the semi-c...

The problem of how to obtain quasi-classical states for quantum groups is examined. A measure of quantum indeterminacy is proposed, which involves expectation values of some natural quantum group operators. It is shown that within any finite dimensional irreducible representation, the highest weight vector and those unitarily related to it are the quasi-classical states. Quantum groups [1] have...

We introduce a notion of “gradient at a given scale” of functions defined on a metric measure space. We then use it to define Sobolev inequalities at large scale and we prove their invariance under large-scale equivalence (maps that generalize the quasi-isometries). We prove that for a Riemmanian manifold satisfying a local Poincaré inequality, our notion of Sobolev inequalities at large scale ...