Liu T Y, Chiu T L, Clarkson P A, Chow K W
Department of Mechanical Engineering, University of Hong Kong, Pokfulam, Hong Kong.
School of Mathematics, Statistics and Actuarial Science, University of Kent, Canterbury, Kent CT2 7FS, United Kingdom.
Chaos. 2017 Sep;27(9):091103. doi: 10.1063/1.5001007.
Rogue waves of evolution systems are displacements which are localized in both space and time. The locations of the points of maximum displacements of the wave profiles may correlate with the trajectories of the poles of the exact solutions from the perspective of complex variables through analytic continuation. More precisely, the location of the maximum height of the rogue wave in laboratory coordinates (real space and time) is conjectured to be equal to the real part of the pole of the exact solution, if the spatial coordinate is allowed to be complex. This feature can be verified readily for the Peregrine breather (lowest order rogue wave) of the nonlinear Schrödinger equation. This connection is further demonstrated numerically here for more complicated scenarios, namely the second order rogue wave of the Boussinesq equation (for bidirectional long waves in shallow water), an asymmetric second order rogue wave for the nonlinear Schrödinger equation (as evolution system for slowly varying wave packets), and a symmetric second order rogue wave of coupled Schrödinger systems. Furthermore, the maximum displacements in physical space occur at a time instant where the trajectories of the poles in the complex plane reverse directions. This property is conjectured to hold for many other systems, and will help to determine the maximum amplitudes of rogue waves.
演化系统中的 rogue 波是在空间和时间上都局域化的位移。从复变函数通过解析延拓的角度来看，波剖面最大位移点的位置可能与精确解极点的轨迹相关。更确切地说，如果允许空间坐标为复数，那么在实验室坐标（实空间和时间）中 rogue 波最大高度的位置被推测等于精确解极点的实部。对于非线性薛定谔方程的 Peregrine 呼吸子（最低阶 rogue 波），这一特性很容易得到验证。本文在此通过数值方法进一步证明了这种联系，针对更复杂的情形，即 Boussinesq 方程（用于浅水中双向长波）的二阶 rogue 波、非线性薛定谔方程（作为缓慢变化波包的演化系统）的非对称二阶 rogue 波以及耦合薛定谔系统的对称二阶 rogue 波。此外，物理空间中的最大位移出现在复平面中极点轨迹反向的时刻。据推测，这一特性对许多其他系统也成立，并且将有助于确定 rogue 波的最大振幅。
