Backlund变换、Ward孤子和么正子
Backlund transformations, Ward solitons, and unitons
摘要：
Ward方程，也称为修正的2 + 1手征模型，是由$R^{2,2}$上的自对偶Yang-Mills场方程通过降维和规范固定得到的.它有一个Lax对，是一个可积系统。Ward构造了孤子，其扩展解具有不同的单极点。他还使用了限制的方法来构建2-孤子的扩展解决方案有一个双极点。Ioannidou和Zakrzewski和Anand构造了更多的孤子解，其扩展解具有双极点或三极点。本文的一些主要结果是：（i）我们构造代数Bäcklund变换（BT），生成新的解决方案的沃德方程从一个给定的代数方法。(ii)我们利用一阶$k$极限方法和代数BT构造了显式Ward孤子，其扩展解具有任意极点和多重性。(iii)我们证明了我们的构造明确给出了Ward方程的所有孤子，并且Ward孤子的元必须是$x, y$和$t$中的有理函数. (iv)由于定态Ward孤子是幺正，我们的方法也给出了从$C$到$C^n$的200个有理映射的所有$k$-幺正的显式构造.
英文摘要：
The Ward equation, also called the modified 2+1 chiral model, is obtained by a dimension reduction and a gauge fixing from the self-dual Yang-Mills field equation on $R^{2,2}$. It has a Lax pair and is an integrable system. Ward constructed solitons whose extended solutions have distinct simple poles. He also used a limiting method to construct 2-solitons whose extended solutions have a double pole. Ioannidou and Zakrzewski, and Anand constructed more soliton solutions whose extended solutions have a double or triple pole. Some of the main results of this paper are: (i) We construct algebraic Bäcklund transformations (BTs) that generate new solutions of the Ward equation from a given one by an algebraic method. (ii) We use an order $k$ limiting method and algebraic BTs to construct explicit Ward solitons, whose extended solutions have arbitrary poles and multiplicities. (iii) We prove that our construction gives all solitons of the Ward equation explicitly and the entries of Ward solitons must be rational functions in $x, y$ and $t$. (iv) Since stationary Ward solitons are unitons, our method also gives an explicit construction of all $k$-unitons from finitely many rational maps from $C$ to $C^n$.
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