论文标题
对于非零边界条件的四阶分散非线性schrödinger方程的长期渐近造型
Long-time asymptotics for a fourth-order dispersive nonlinear Schrödinger equation with nonzero boundary conditions
论文作者
论文摘要
储层计算是预测湍流的有力工具，其简单的架构具有处理大型系统的计算效率。然而，其实现通常需要完整的状态向量测量和系统非线性知识。我们使用非线性投影函数将系统测量扩展到高维空间，然后将其输入到储层中以获得预测。我们展示了这种储层计算网络在时空混沌系统上的应用，该系统模拟了湍流的若干特征。我们表明，使用径向基函数作为非线性投影器，即使只有部分观测并且不知道控制方程，也能稳健地捕捉复杂的系统非线性。最后，我们表明，当测量稀疏、不完整且带有噪声，甚至控制方程变得不准确时，我们的网络仍然可以产生相当准确的预测，从而为实际湍流系统的无模型预测铺平了道路。
In this work, we consider the long-time asymptotics for the Cauchy problem of a fourth-order dispersive nonlinear Schrödinger equation with nonzero boundary conditions at infinity. Firstly, in order to construct the basic Riemann-Hilbert problem associated with nonzero boundary conditions, we analysis direct scattering problem. Then we deform the corresponding matrix Riemann-Hilbert problem to explicitly solving models via using the nonlinear steepest descent method and employing the $g$-function mechanism to eliminate the exponential growths of the jump matrices. Finally, we obtain the asymptotic stage of modulation instability for the fourth-order dispersive nonlinear Schrödinger equation.
