Kaup-Broer系统的解及其2+1维可积推广的Dressing方法
Solutions to the Kaup--Broer System and Its 2+1 Dimensional Integrable Generalization via the Dressing Method
摘要：
本文对1+1维Kaup-Broer系统[7，13]的4个标度类，给出了Manakov和Zakharov [28，29，27]的非局部dbar问题dressing方法. 1+1维Kaup-Broer方程组的方法是文献[23]中首次提出的复值2+1维完全可积偏微分方程的方法的简化。这种方法允许计算的解决方案的所有情况下的Kaup-Broer系统。然后，我们考虑的情况下，通常的引力强迫的非毛细波，并使用敷料的方法来计算N-孤立子的解决方案和更一般的解决方案，在封闭的N-孤立子解决方案的拓扑结构中的一致收敛的紧集称为原始的解决方案。这些更一般的解类似于[11，30，31]中KdV方程的解。我们推导出有限间隙解的修整函数。我们计算对向传播的色散冲击波型解的数值。
英文摘要：
In this paper we formulate the nonlocal dbar problem dressing method of Manakov and Zakharov [28, 29, 27] for the 4 scaling classes of the 1+1 dimensional Kaup--Broer system [7, 13]. The method for the 1+1 dimensional Kaup--Broer systems are reductions of a method for a complex valued 2+1 dimensional completely integrable partial differential equation first introduced in [23]. This method allows computation of solutions to all cases of the Kaup--Broer system. We then consider the case of non-capillary waves with usual gravitational forcing, and use the dressing method to compute N-soliton solutions and more general solutions in the closure of the N-soliton solutions in the topology of uniform convergence in compact sets called primitive solutions. These more general solutions are an analogue of the solutions derived in [11, 30, 31] for the KdV equation. We derive dressing functions for finite gap solutions. We compute counter propagating dispersive shockwave type solutions numerically.
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