Enhancement of the Zakharov-Glassey's method for Blow-Up in nonlinear Schroedinger equations

薛定谔方程的钟摆观

Pendulum View of The Schroedinger Equation

在一维薛定谔方程和单摆问题之间找到了一个变换。证明了如何构造精确解的摆方程。指出了这种变换与Zakharov-Shabat方程的关系。

Stability and cascades for the Kolmogorov-Zakharov spectrum of wave turbulence

The kinetic wave equation arises in wave turbulence to describe the Fourier spectrum of solutions to the cubic Schroedinger equation. The equation has two Kolmogorov-Zakharov steady states corresponding to out-of-equilibrium cascades transferring for the first solution mass from infinity to zero (small spatial scales to large scales) and for the other solution energy from zero to infinity. After conjecturing the generic development of the two cascades, we verify it partially in the isotropic case by proving the nonlinear stability of the mass cascade in the stationary setting. This constructs …

格点势KdV方程的多尺度展开和可积性

Multiscale expansion and integrability properties of the lattice potential KdV equation

我们应用离散多尺度展开的拉克斯对和晶格势Korteweg-de Vries方程的前几个对称性。从这些计算中，我们发现，像最低阶的世俗性条件给出一个非线性Schroedinger方程一样，拉克斯对在相同的阶上给出了Zakharov和Shabat谱问题，对称性给出了非线性Schroedinger方程的点族和广义对称性.

Dressing method and quadratic bundles related to symmetric spaces. Vanishing boundary conditions

We consider quadratic bundles related to Hermitian symmetric spaces of the type SU(m+n)/S(U(m)x U(n)). The simplest representative of the corresponding integrable hierarchy is given by a multi-component Kaup-Newell derivative nonlinear Schroedinger equation which serves as a motivational example for our general considerations. We extensively discuss how one can apply Zakharov-Shabat's dressing procedure to derive reflectionless potentials obeying zero boundary conditions. Those could be used for one to construct fast decaying solutions to any nonlinear equation belonging to the same hierarchy.…