Multiple complex soliton solutions for integrable negative-order KdV and integrable negative-order modified KdV equations

In this work we show that the integrable negative-order Korteweg–de Vries (nKdV) and the integrable negative-order modified Korteweg–de Vries (nMKdV) equation admit multiple complex soliton solutions. To achieve this goal, we introduce two complex forms of the simplified Hirota’s method, the first works effectively for the nKdV equation, and the other form is nicely applicable for the nMKdV equation. We believe that the newly proposed complex forms and the obtained findings will shed light on complex solitons of other integrable equations.