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We have derived the extended Korteweg-de Vries equation describing the longgravity waves without limitation to surface deviation. The only restriction tothe surface deviation is connected with the stability condition for appropriatesolutions. The derivation of extended KdV equation is based on the Eulerequations for inviscid irrotational and incompressible fluid. It is shown thatthe extended KdV equation reduces to standard KdV equation for small amplitudeof the waves. We have also generalized the extended KdV equation for describingthe decaying effect of the waves. Quasi-periodic and solitary wave solutionsfor extended KdV equation with decaying effect are found as well. We alsodemonstrate that the fundamental approach based on the inverse scatteringmethod is applicable for solving the extended KdV equation in the case whendecaying effect is negligibly small. Such case always occur for restrictedpropagation distances of the waves.

Extended Korteweg-de Vries equation for long gravity waves in incompressible fluid without strong limitation to surface deviation