论文标题
变形猜想：通过使用保护定律，将较低尺寸的可整合系统变形到更高维度的系统
Deformation Conjecture: Deforming Lower Dimensional Integrable Systems to Higher Dimensional Ones by Using Conservation Laws
论文作者
论文摘要
利用（1+1） - 维局部局部进化系统的一些保护定律，可以猜想较高的尺寸可集成方程可以通过变形算法定期构建。该算法可以应用于松弛对和高阶流量。换句话说，如果原始的较低维模型是可松弛的集成（具有松弛对）和对称性集成（具有无限的高阶对称性），则变形的高阶系统也是可宽松的集成和对称性集成的。对于具体性，将变形算法应用于常规（1+1）二维KDV方程和（1+1）-Dirensional AKNS系统（包括非线性NLS方程作为一个特殊示例）。有趣的是，变形（3+1）维kDV方程也是（1+1） - 维harry-dym（HD）类型方程的扩展，这是（1+1）-Dimensional KDV方程的相互链接。明确给出了（3+1） - 维kDV-HD系统和（2+1） - 维AKNS系统的宽松对。高阶对称性，即通过变形算法明确获得了整个（3+1）二维KDV-HD层次结构。 （3+1） - 维kDv-hd方程的单齿溶液被隐式地给出。由于变形的影响，通常的KDV方程的对称孤子形状不再保守，并且变形为不对称和/或多价。对于几乎所有已知的（1+1）二维整合局部进化系统，变形猜想是正确的，到目前为止，我们尚未发现任何反例。引入了大量（D+1） - 二维整合系统的本文探讨了所有数学家和理论物理学家的严重挑战，因为传统方法不再直接有效地解决这些集成方程。
Utilizing some conservation laws of (1+1)-dimensional integrable local evolution systems, it is conjectured that higher dimensional integrable equations may be regularly constructed by a deformation algorithm. The algorithm can be applied to Lax pairs and higher order flows. In other words, if the original lower dimensional model is Lax integrable (possesses Lax pairs) and symmetry integrable (possesses infinitely many higher order symmetries), then the deformed higher order systems are also Lax integrable and symmetry integrable. For concreteness, the deformation algorithm is applied to the usual (1+1)-dimensional KdV equation and the (1+1)-dimensional AKNS system (including nonlinear NLS equation as a special example). It is interesting that the deformed (3+1)-dimensional KdV equation is also an extension of the (1+1)-dimensional Harry-Dym (HD) type equations which are reciprocal links of the (1+1)-dimensional KdV equation. The Lax pairs of the (3+1)-dimensional KdV-HD system and the (2+1)-dimensional AKNS system are explicitly given. The higher order symmetries, i.e., the whole (3+1)-dimensional KdV-HD hierarchy, are also explicitly obtained via the deformation algorithm. The single soliton solution of the (3+1)-dimensional KdV-HD equation is implicitly given. Because of the effects of the deformation, the symmetric soliton shape of the usual KdV equation is no longer conserved and deformed to be asymmetric and/or multi-valued. The deformation conjecture is correct for almost all the known (1+1)-dimensional integrable local evolution systems and we have not yet found any counter-example so far. The introduction of a large number of (D+1)-dimensional integrable systems of this paper explores a serious challenge to all mathematicians and theoretical physicists because the traditional methods are no longer directly valid to solve these integrable equations.
