Whitham modulation theory for the defocusing nonlinear Schrödinger equation in two and three spatial dimensions
Whitham modulation theory for the defocusing nonlinear Schrödinger equation in two and three spatial dimensions
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二维和三维空间中散焦非线性薛定谔方程的 Whitham 调制理论
DOI：
10.1088/1751-8121/acb117
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发表时间：
2023
期刊：
影响因子：
--
通讯作者：
Hoefer, Mark A
中科院分区：
文献类型：
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作者：
Abeya, Asela;Biondini, Gino;Hoefer, Mark A
利用周期行波解的两相近似和对非线性薛定谔方程的守恒律进行周期平均，导出了二维、三维和更高维空间中的非线性薛定谔方程的Whitham调制方程.由此产生的Whitham调制方程以矢量形式写成，这使得人们可以证明它们保留了NLS方程的旋转不变性，以及相对于缩放和伽利略变换的不变性，并立即将计算从二维空间推广到三维空间。黎曼型变量的变换进行了详细描述;明确写下来的Whitham调制方程的谐波和孤子的限制;和减少的Whitham方程的径向NLS方程明确进行。最后，扩展的理论，以更高的空间维度简要概述。多维NLS-Whitham方程可以用来研究在各种应用中的大振幅波列，包括非线性光子学和物质波。
The Whitham modulation equations for the defocusing nonlinear Schrödinger (NLS) equation in two, three and higher spatial dimensions are derived using a two-phase ansatz for the periodic traveling wave solutions and by period-averaging the conservation laws of the NLS equation. The resulting Whitham modulation equations are written in vector form, which allows one to show that they preserve the rotational invariance of the NLS equation, as well as the invariance with respect to scaling and Galilean transformations, and to immediately generalize the calculations from two spatial dimensions to three. The transformation to Riemann-type variables is described in detail; the harmonic and soliton limits of the Whitham modulation equations are explicitly written down; and the reduction of the Whitham equations to those for the radial NLS equation is explicitly carried out. Finally, the extension of the theory to higher spatial dimensions is briefly outlined. The multidimensional NLS-Whitham equations obtained here may be used to study large amplitude wavetrains in a variety of applications including nonlinear photonics and matter waves.