Non existence of L2–compact solutions of the Kadomtsev–Petviashvili II equation
Non existence of L2–compact solutions of the Kadomtsev–Petviashvili II equation
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Kadomtsev-Petviashvili II 方程不存在 L2 紧致解
DOI：
10.1007/s00208-003-0498-6
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发表时间：
2004
期刊：
影响因子：
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通讯作者：
Y. Martel
中科院分区：
文献类型：
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作者：
A. de Bouard;Y. Martel
证明了Kadomtsev-Petviashvili II方程（KP II方程）$${{ (u_t+u_{{xxx}}+uu_x)_x+u_{{yy}}=0,\quad (x,y)\in {{\bf{ R}}}^2, }}$$在x变量中不存在2紧化（即在2范数中一致定域）且向右行进的非平凡解。这一结果推广了de Bouard和Saut[3]先前关于KP II方程不存在行波解的工作。证明使用了KP II方程（类似于KdV方程[12]，[14]）和两个维里型关系的解的质量单调性。该结果仍然适用于KP II方程的一些自然推广（一般非线性，高色散），并且不依赖于方程的可积性。
We prove that there is no nontrivial solution of the Kadomtsev–Petviashvili II equation (KP II equation) $${{ (u_t+u_{{xxx}}+uu_x)_x+u_{{yy}}=0,\quad (x,y)\in {{\bf{ R}}}^2, }}$$ which isL2compact (i.e. uniformly localized inL2norm) and travel to the right in thexvariable. This result extends the previous work of de Bouard and Saut [3] stating that there is no traveling wave solution for the KP II equation. The proof uses a monotonicity property of theL2mass for solutions of the KP II equation (similar to the one for the KdV equation [12], [14]) and two virial type relations. The result still holds for some natural generalizations of the KP II equation (general nonlinearity, higher dispersion) and does not rely on the integrability of the equation.