论文标题
几何整合方程的结构和定性特性
Structural and qualitative properties of a geometrically integrable equation
论文作者
论文摘要
找到了诺维科夫几何积分方程的谎言对称性，并获得了群体不变的溶液。建立了二阶的地方保护法及其相应的保守量。提出了$ l^1 $规范不变的足够条件，以及存在正解决方案的条件。给出了两个用于唯一延续解决方案的演示：其中之一只是基于解决方案的$ l^1 $规范的不变性，而另一个是基于库奇问题的适应性。最后，研究了由方程溶液确定的伪球面：所有不导致伪球体表面的不变溶液进行了分类，并且使用溶液和供应良好的结果证明了用于伪球面的分析度量标准的存在。
Lie symmetries of a Novikov geometrically integrable equation are found and group-invariant solutions are obtained. Local conservation laws up to second order are established as well as their corresponding conserved quantities. Sufficient conditions for the $L^1$ norm of the solutions to be invariant are presented, as well as conditions for the existence of positive solutions. Two demonstrations for unique continuation of solutions are given: one of them is just based on the invariance of the $L^1$ norm of the solutions, whereas the other is based on well-posedness of Cauchy problems. Finally, pseudo-spherical surfaces determined by the solutions of the equation are studied: all invariant solutions that do not lead to pseudo-spherical surfaces are classified and the existence of an analytic metric for a pseudo-spherical surface is proved using conservation of solutions and well-posedness results.
