(2+1)维Zakharov方程的自相似变换和线怪波簇激发

Self-similar transformation and excitation of rogue waves for(2+1)-dimensional Zakharov equation

首先建立(2+1)维(二维空间和一维时间)Zakharov方程的自相似变换,并将该系统转换为(1+1)维非线性薛定谔(nonlinear Schr?dinger, NLS)方程;然后基于该相似变换和已知的(1+1)维NLS方程有理形式解,通过选择合适参数得到了(2+1)维Zakharov方程在x-y平面上丰富的线怪波簇激发,发现产生线怪波簇最大辐值时的传播距离z值完全不同,而且形状和幅度可以得到有效调控;最后借助图示展现了二维怪波的传播特征.此外,发现在x-y平面上,当参数γ=1时,呈现线怪波;而当参数γ/=1时,线怪波转变为离散的局域怪波.随参数γ的增大,可以在x-y平面限定区域获得时空局域的怪波,这与Peregrine在(1+1)维NLS方程中发现的“Kuznetsov-Ma孤子”(Kuznetsov-Masoliton,KMS)或“Akhmediev呼吸子”(Akhmedievbreather,AB)极限情形的“Peregrine孤子”(Peregrine soliton, PS)类似.本文提出的(2+1)维Zakharov方程怪波方法可以作为获得高维怪波激发的有效途径,并推广应用于其他(2+1)维非线性系统.

The search for the excitation of two-dimensional rogue wave in a(2+1)-dimensional nonlinear evolution model is a research hotspot. In this paper, the self-similar transformation of the(2+1)-dimensional Zakharov equation is established, and this equation is transformed into the(1+1)-dimensional nonlinear Schr?dinger equation. Based on the similarity transformation and the rational formal solution of the(1+1)-dimensional nonlinear Schr?dinger equation, the rogue wave excitation of the(2+1)-dimensional Zakharov equation is obtained by selecting appropriate parameters. We can see that the shape and amplitude of the rogue waves can be effectively controlled. Finally, the propagation characteristics of line rogue waves are diagrammed visually.We also find that the line-type characteristics of two-dimensional rogue wave are present in the x-y plane when the parameter γ = 1. The line rogue wave is converted into discrete localized rogue wave in the x-y plane when the parameter γ =/1. The spatial localized rogue waves with short-life can be obtained in the required x-y plane region. This is similar to the Peregrine soliton(PS) first discovered by Peregrine in the(1+1)-dimensional NLS equation, which is the limit case of the “Kuznetsov-Ma soliton”(KMS) or “Akhmediev breather”(AB). The proposed approach to constructing the line rogue waves of the(2+1) dimensional Zakharov equation can serve as a potential physical mechanism to excite two-dimensional rogue waves, and can be extended to other(2+1)-dimensional nonlinear systems.