Soliton–mean field interaction in Korteweg–de Vries dispersive hydrodynamics
Soliton–mean field interaction in Korteweg–de Vries dispersive hydrodynamics
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DOI：
10.1111/sapm.12615
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发表时间：
2022-11
影响因子：
2.7
通讯作者：
M. Ablowitz;J. Cole;G. El;M. Hoefer;Xu‐Dan Luo
中科院分区：
文献类型：
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作者：
M. Ablowitz;J. Cole;G. El;M. Hoefer;Xu‐Dan Luo
在大尺度波的存在下局域孤子的数学描述是非线性科学中的一个基本问题，在流体动力学，非线性光学和凝聚态物理学中有应用。在这里，孤立子的演化，因为它与稀疏波或色散冲击波，缓慢变化和快速振荡的色散平均场的例子，Korteweg-de弗里斯方程的相互作用进行了研究。阶跃边界条件产生稀疏波（阶跃上升）或色散激波（阶跃下降）。当孤子与这些平均场相互作用时，它可以通过（隧道）或嵌入（捕获）其中，这取决于它的初始振幅和位置。三个独立的分析方法进行专题审查，以描述这些相互作用。首先，介绍了一个基本的孤子微扰理论，发现捕获的孤子稀疏波相互作用的小色散限制的解决方案的动力学。其次，多相Whitham调制理论及其有限间隙描述被用来描述孤子-稀疏波和孤子-色散激波相互作用。最后，通过逆散射变换得到了初值问题的谱描述和精确解。对于传输孤子，远场渐近性揭示了通过上述任何类型的波的孤子相移。在陷波情况下，在谱描述中没有适当的本征值，这意味着演化不涉及适当的孤子解。这些方法是一致的，同意直接数值模拟，并准确地描述孤立波平均场相互作用的不同方面。
The mathematical description of localized solitons in the presence of large‐scale waves is a fundamental problem in nonlinear science, with applications in fluid dynamics, nonlinear optics, and condensed matter physics. Here, the evolution of a soliton as it interacts with a rarefaction wave or a dispersive shock wave, examples of slowly varying and rapidly oscillating dispersive mean fields, for the Korteweg–de Vries equation is studied. Step boundary conditions give rise to either a rarefaction wave (step up) or a dispersive shock wave (step down). When a soliton interacts with one of these mean fields, it can either transmit through (tunnel) or become embedded (trapped) inside, depending on its initial amplitude and position. A topical review of three separate analytical approaches is undertaken to describe these interactions. First, a basic soliton perturbation theory is introduced that is found to capture the solution dynamics for soliton–rarefaction wave interaction in the small dispersion limit. Next, multiphase Whitham modulation theory and its finite‐gap description are used to describe soliton–rarefaction wave and soliton–dispersive shock wave interactions. Lastly, a spectral description and an exact solution of the initial value problem is obtained through the inverse scattering transform. For transmitted solitons, far‐field asymptotics reveal the soliton phase shift through either type of wave mentioned above. In the trapped case, there is no proper eigenvalue in the spectral description, implying that the evolution does not involve a proper soliton solution. These approaches are consistent, agree with direct numerical simulation, and accurately describe different aspects of solitary wave–mean field interaction.