Resonant collisions among localized lumps and line solitons of the Kadomtsev–Petviashvili I (KP-I) equation are studied. The KP-I equation describes the evolution of weakly nonlinear, weakly dispersive waves with slow transverse variations. Lumps can only exist for the KP equation when the signs of the transverse derivative and the weak dispersion in the propagation direction are different, that is, in the KP-I regime. Collisions among lumps and solitons for “integrable” equations are normally elastic, that is, wave shapes are preserved except possibly for phase shifts. For resonant collisions, mathematically the phase shift will become indefinitely large. Physically a lump may be detached (or emitted) from a line soliton, survives for a brief transient period in time, and then merges with the next adjacent soliton. This special lump is thus localized in time as well as the two spatial dimensions, and can be termed a “rogue lump.” By employing a reduction method for the KP hierarchy in conjunction with the Hirota bilinear technique, a general -lump/-line-soliton solution is obtained for the KP-I equation for an arbitrary positive integer . For , collisions among modes become increasingly complicated, with multiple lumps detaching from one single or different line solitons and then disappearing into the remaining solitons. In terms of applications, such “rogue lumps” represent a mode truly localized both in space and time, and will be valuable in modeling physical problems.

中文翻译：

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Kadomtsev-Petviashvili I 方程中块和线孤子的完全共振碰撞

研究了 Kadomtsev-Petviashvili I (KP-I) 方程的局部团块和线孤子之间的共振碰撞。KP-I 方程描述了具有缓慢横向变化的弱非线性、弱色散波的演变。只有当横向导数的符号和传播方向上的弱色散的符号不同时，即在 KP-I 状态下，KP 方程才能存在团块。对于“可积”方程，块体和孤子之间的碰撞通常是弹性的，也就是说，除了相移可能之外，波形都保持不变。对于共振碰撞，数学上的相移将无限大。从物理上讲，一个团块可能会从一个线孤子上分离（或发射），在短时间内存活一段时间，然后与下一个相邻的孤子合并。因此，这种特殊的肿块在时间和两个空间维度上都被定位，可以称为“流氓肿块”。通过结合 Hirota 双线性技术对 KP 层次结构采用归约方法，一般的-lump/ -line-soliton 解是针对任意正整数的 KP-I 方程获得的。对于，模式之间的碰撞变得越来越复杂，多个团块从一个单独的或不同的线孤子中分离出来，然后消失在剩余的孤子中。在应用方面，这种“流氓块”代表了一种在空间和时间上都真正局部化的模式，并且在建模物理问题方面非常有价值。