New reductions of the Kadomtsev–Petviashvili and two‐dimensional Toda lattice hierarchies via symmetry constraints
New reductions of the Kadomtsev–Petviashvili and two‐dimensional Toda lattice hierarchies via symmetry constraints
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DOI：
10.1063/1.529862
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发表时间：
1992-11
影响因子：
1.3
通讯作者：
B. Konopelchenko;W. Strampp
中科院分区：
文献类型：
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作者：
B. Konopelchenko;W. Strampp
在Sato方法的基础上，研究了Kadomtsev-Petviashvili（KP）和二维户田格（2DTL）的新的约化.在这种方法中，这些层次由势u1，u2，u3，.的无限组方程表示，伪微分算子及其本征函数和伴随本征函数。本文研究了KP族和2DTL族在下列类型的约束下的解：∑n= 1 N αnSn（u1，u2，u3，...）=Ωx，其中Sn是这些族的对称性，αn是任意常数，Ω是数量<$（λ）<$*（μ）的任意线性泛函。结果表明，对于KP层次，这些约束会导致变量u2，u3，.，uN，uN，uN *。许多已知的系统和一些新的系统都在其中。2DTL族的对称约化同时产生有限维动力系统和1+1维离散系统。对修正的KP方程组给出了一些结果。
New types of reductions of the Kadomtsev–Petviashvili (KP) hierarchy and the two‐dimensional Toda lattice (2DTL) hierarchy are considered on the basis of Sato’s approach. Within this approach these hierarchies are represented by infinite sets of equations for potentials u1,u2,u3,..., of pseudodifferential operators and their eigenfunctions ψ and adjoint eigenfunctions ψ*. The KP and the 2DTL hierarchies are studied under constraints of the following type: ∑n=1N αnSn(u1,u2,u3,...)=Ωx, where Sn are symmetries for these hierarchies, αn are arbitrary constants, and Ω is an arbitrary linear functional of the quantity ψ(λ)ψ*(μ). It is shown that for the KP hierarchy these constraints give rise to hierarchies of 1+1‐dimensional commuting flows for the variables u2,u3,...,uN,ψ,ψ*. Many known systems and several new ones are among them. Symmetry reductions for the 2DTL hierarchy give rise both to finite‐dimensional dynamical systems and 1+1‐dimensional discrete systems. Some few results for the modified KP hierarc...