标题： 负Korteweg-de Vries（nKdV）方程和负修正Korteweg-de Vries（nmKdV）方程的精确移动解与1+1维$φ^4$场论静态解之间的联系 显示英文标题

标题： Connection Between the Exact Moving Solutions of the Negative Korteweg-de Vries (nKdV) Equation and the Negative Modified Korteweg-de Vries (nmKdV) Equation and the Static Solutions of 1+1 Dimensional $φ^4$ Field Theory

摘要： 负阶KdV（nKdV）和修正KdV（nmKdV）方程基于不同的层次算子有两种不同的表述形式。 这两个方程都可以表示为关于场$u(x,t)$的非线性微分方程，我们称这种形式为方程的“Lou形式”。 我们发现，对于用$u(x,t) \rightarrow u (x-ct)= u(\xi) $表示的nKdV方程和nmKdV方程的移动解，在“Lou形式”下，$u(\xi)$方程可以映射到1+1维$\phi^4$场理论静态解的方程。 利用这种映射，我们得到了大量nKdV和nmKdV方程的解，其中大部分是新的。 我们还证明了nKdV方程可以从它的两种表述形式中的作用量原理导出。 此外，对于nmKdV方程的两种形式以及聚焦和去聚焦情况，我们证明了通过合适的假设，可以分离nmKdV场$u(x,t)$的$x$和$t$依赖关系，并在所有情况下获得新的解。 我们还获得了nKdV和nmKdV方程的新有理解。 显示英文摘要

摘要： The negative order KdV (nKdV) and the modified KdV (nmKdV) equations have two different formulations based on different hierarchy operators. Both equations can be written in terms of a nonlinear differential equation for a field $u(x,t)$ which we call the ``Lou form" of the equation. We find that for moving solutions of the nKdV equation and the nmKdV equation written in the ``Lou form" with $u(x,t) \rightarrow u (x-ct)= u(\xi) $, the equation for $u(\xi)$ can be mapped to the equation for the static solutions of the 1+1 dimensional $\phi^4$ field theory. Using this mapping we obtain a large number of solutions of the nKdV and the nmKdV equation, most of which are new. We also show that the nKdV equation can be derived from an Action Principle for both of its formulations. Furthermore, for both forms of the nmKdV equations as well as for both focusing and defocusing cases, we show that with a suitable ansatz one can decouple the $x$ and $t$ dependence of the nmKdV field $u(x,t)$ and obtain novel solutions in all the cases. We also obtain novel rational solutions of both the nKdV and the nmKdV equations.