Simple Explode-Decay Mode Solution of a Certain One-Space-Dimensional Nonlinear Evolution Equation

We study the so-called completely integrable nonlinear evolution equation. Recently it has been found that certain two-space-dimensional nonlinear evolution equations have not only well-known soliton solutions but also simple (similarity type) explode-decay mode solutions expressed by a closed form of simple analytic functions. In the one-space-dimensional system, so far, there have been no known example which admit simple explode-decay mode solutions as well as solitons. We have found one explicit example which has such property.