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A rigorous derivation of the extended KdV equation

The interesting background and historical development of KdV equations were discussed widely. These equations describe the propagation of water waves in weakly non linear and weakly dispersive medium. Referring to physical derivation of KdV equations, scientists used to impose shallow water equations, thus the formal or physical derivation of KdV equations. However, these equations have rarely been derived rigorously. The aim of this paper is to giving insight into their rigorous mathematical derivation, instead of only referring to. Thereby, a rigorous derivation of two extended KdV equations: one on the velocity, other on the surface elevation. With this aim in mind, the primary research method for this paper will depend on the definition of consistency. Hence, a rigorous justification of new extended KdV equations will be provided thanks to this definition. This result provides a precise mathematical answer to a question raised by several authors in the last years, that is the verification of the extended KdV equations, derived previously, using formal methods.


Introduction
Over the decades, the struggle over energy resources has not stopped. Now, it is universally accepted that fossil fuels are finite, and with the passage of time it will be exhausted. Since it is a matter of time, scientists turned into finding new energy resources. Known as one of the world's most abundant sources of renewable energy, ocean energy became an innovative solution for new energy challenges. Taking into consideration the huge amount of energy carried by ocean waves across the sea, many different techniques for converting wave energy to electric power have been investigated. However, generated as a result of wind blowing over the ocean surface, ocean energy is the most concentrated form of renewable energy on earth, also it is more predictable and consistent than wind or solar energy. Due to the difficulty and complexity of water waves problems, mathematicians, physicists and oceanographers get to find new asymptotic models with the same accuracy as the original ones, like Green-Naghdi and Boussinessq systems. Thence, the definition of consistency serves mathematicians in the resolution of these models, by deriving consistent equations. One of these non linear equations is KdV equation, that is the subject of concentrated study to understand the physical phenomena of oceanography. Korteweg and his student de Vries derived their non linear wave equation to describe the shallow water waves that Russell had observed in 1834. The most renowned KdV equation is [5]: that was originally derived for flat bottom. What attracted the focus of scientists in this equation, was its integrability property, and thus its solitons (solitary waves) solutions. Since the derivation of the equation mentioned before, several methods have been used to derive new extended KdV equations, with different ocean conditions and properties. General derivations of this equation were justified with bottom, and with non constant coefficients, with topography (called KdV-top equation) [5,6]. A formal derivation of KdV equation was provided using Whitham method, in the presence of surface tension [2], all these previous works were done up to O(µ 2 ). Also, using some physical principles, an extended KdV equation was formally derived up to O(µ 3 ) [8].
In the paper at hand, we deal with an irrotational, incompressible, inviscid fluid with a free surface, and constant density, acted on only by gravity. Knowing that a is the amplitude of the wave, λ is the wave-length of the wave, h 0 is the reference depth, denote by where u is the fluid velocity, ζ is the surface elevation, µ = h 0 2 λ 2 and ε = a h 0 are the shallowness and nonlinearity parameters respectively. Recall that the KdV scaling is ε = O(µ), with 0 < ε ≤ 1 and 0 < µ 1.
The organization of this paper is as follows: in the second section, a derivation of the extended KdV equation will be done, in the first subsection 2.1, we will derive rigorously an extended KdV equation on the velocity u. In the second subsection 2.2, a rigorous mathematical derivation of extended KdV equation on the surface elevation ζ will be provided, and hence a rigorous verification of this imposed  [8]. The aim of this paper is to give a rigorous mathematical derivation of the extended KdV equations, up to µ 3 . Concerning the methodolgy, after the examination of some previous works, we will proceed as in [1], so we will use the definition of consistency to provide these rigorous derivations, which guarantee the relevance of these equations with (1), and serve in the construction of approximate solutions of Boussinesq equations.

Derivation of new extended KdV equations
The main goal of this section is to find extended KdV equations on velocity and on surface elevation.
The new derived KdV equation on the velocity will be: And one on the surface elevation will be:

Equation on the velocity.
In order to get (2), we will introduce the following equation on u where α, β, γ, δ are parameters in R. Next, we need to find the values of parameters mentioned above, that will be done in three steps.
• Step1: From (4) we get: Substitute the expression of u xxx in (4) to get: One can get To find another equation on a, b, c and α we need to use the equations of (1).
In the next step, the second equation of (1) we will be used.
• Step2: Let v be a smooth enough function such that ζ = u + εv. Then (1) 2 becomes We know from (5) that Now, we will use the first equation of (1). Put ζ = u + εv in (1) 1 that is Multiply (6) by εu, derive with respect to x and use (7) to get Next, deriving (6) with repect to t one can get Also we have Gathering all the computations done above one can get Substitute ε 2 (uv) x and (εv) t in (1) 1 to get Compare the equations (5) and (8)

Equation on the surface elevation.
In this subsection, we are going to derive the extended KdV equation on the surface elevation ζ as a rigorous verification of a previous work [8], where it was imposed formally. For this purpose, let us introduce the following equation where α, β, γ, δ are parameters in R, and will be determined next.