Kedziora David J, Ankiewicz Adrian, Akhmediev Nail
Optical Sciences Group, Research School of Physics and Engineering, The Australian National University, Canberra ACT 0200, Australia.
Phys Rev E Stat Nonlin Soft Matter Phys. 2012 Jun;85(6 Pt 2):066601. doi: 10.1103/PhysRevE.85.066601. Epub 2012 Jun 4.
We present an explicit analytic form for the two-breather solution of the nonlinear Schrödinger equation with imaginary eigenvalues. It describes various nonlinear combinations of Akhmediev breathers and Kuznetsov-Ma solitons. The degenerate case, when the two eigenvalues coincide, is quite involved. The standard inverse scattering technique does not generally provide an answer to this scenario. We show here that the solution can still be found as a special limit of the general second-order expression and appears as a mixture of polynomials with trigonometric and hyperbolic functions. A further restriction of this particular case, where the two eigenvalues are equal to i, produces the second-order rogue wave with two free parameters considered as differential shifts. The illustrations reveal a precarious dependence of wave profile on the degenerate eigenvalues and differential shifts. Thus we establish a hierarchy of second-order solutions, revealing the interrelated nature of the general case, the rogue wave, and the degenerate breathers.
我们给出了具有虚特征值的非线性薛定谔方程的双呼吸子解的显式解析形式。它描述了阿赫梅迪耶夫呼吸子和库兹涅佐夫 - 马孤子的各种非线性组合。当两个特征值重合时的简并情况相当复杂。标准的逆散射技术通常无法解决这种情况。我们在此表明，该解仍可作为一般二阶表达式的特殊极限找到，并且表现为多项式与三角函数和双曲函数的混合。这种特殊情况的进一步限制，即两个特征值等于i，产生了具有两个作为微分位移的自由参数的二阶 rogue 波。这些图示揭示了波剖面与简并特征值和微分位移之间不稳定的依赖关系。因此，我们建立了二阶解的层次结构，揭示了一般情况、rogue 波和简并呼吸子之间的相互关联性质。
