Infinitely extended complex KdV equation and its solutions: solitons and rogue waves

We present an infinitely-extended KdV equation that contains an infinite number of arbitrary real coefficients controlling higher-order terms in the extended evolution equation. The higher-order terms are chosen in a way that maintains the integrability of the whole equation. Another significant step in this work is that this extended equation admits complex-valued solutions. This generalization allows us to consider both solitons and rogue waves in the form of rational solutions of this equation. Special choices of the arbitrary coefficients lead to particular cases - the basic KdV and its higher-order versions. Using the extended KdV, instead of the basic one, may improve the accuracy of the description of rogue waves in shallow water.