Korteweg-de Vries方程孤子凝聚体的色散流体力学
Dispersive hydrodynamics of soliton condensates for the Korteweg-de Vries equation
AI总结：
本文证明KdV方程稠密孤子气体在凝聚极限下的动力学方程约化为$N$相KdV-Whitham调制方程，并构造Riemann问题显式解及数值揭示稀释凝聚体的可积湍流行为。
摘要：
我们研究Korteweg-de Vries（KdV）方程在特殊“凝聚”极限下非平衡稠密孤子气体的大尺度动力学。我们证明，在此极限下，关于谱态密度的积分-微分动力学方程可约化为由Flaschka、Forest和McLaughlin（1980）以及Lax和Levermore（1983）导出的$N$相KdV-Whitham调制方程。我们考虑孤子凝聚体的Riemann问题，并构造描述广义稀疏波和色散冲击波的动力学方程显式解。随后，我们给出“稀释”孤子凝聚体的数值结果，这些凝聚体表现出与可积湍流相关的丰富非相干行为。
英文摘要：
We consider large-scale dynamics of non-equilibrium dense soliton gas for the Korteweg-de Vries (KdV) equation in the special "condensate" limit. We prove that in this limit the integro-differential kinetic equation for the spectral density of states reduces to the $N$-phase KdV-Whitham modulation equations derived by Flaschka, Forest and McLaughlin (1980) and Lax and Levermore (1983). We consider Riemann problems for soliton condensates and construct explicit solutions of the kinetic equation describing generalized rarefaction and dispersive shock waves. We then present numerical results for "diluted" soliton condensates exhibiting rich incoherent behaviours associated with integrable turbulence.
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