Generic Equations for Long Gravity Waves in Incompressible Fluid with Finite Amplitude

We present the derivation of generic equations describing the long gravity waves in incompressible fluid with a decaying effect. We show that in this theory, the only restriction to the surface deviation is connected to the stability condition for the waves. Derivation of these generic equations is based on Euler equations for inviscid incompressible fluid and the definition of dynamic pressure which leads to a correct dispersion equation for gravity waves. These derived generic equations for the velocity of fluid and the surface deviation describe the propagation of long gravity waves in incompressible fluid with finite amplitude. We also find the necessary and sufficient conditions for generic equations with dissipation of energy or a decaying effect. The developed approach can significantly improve the accuracy of theory for long gravity waves in incompressible fluid. We also find the quasi-periodic and solitary wave solutions for generic equations with a decaying effect.