Toufic El Arwadi
email: t.elarwadi@bau.edu.lb

Samer Israwi
email: sisrawi83@hotmail.com

Samer Isarwi

Toufic El Arwadi

AN EXPLICIT SOLUTION WITH CORRECTORS FOR VARIABLE DEPTH KDV AND CAMASSA-HOLM-LIKE EQUATIONS

. We derive here new asymptotic models of these models so that they have the same accuracy as the standard equations. We solve explicitly the new linear models and numerically validate the results and then we prove numerically the property of breaking wave for the Camassa-Holm equation with non flat bottom.

1. Introduction 1.1. General setting. This paper deals with the water waves problem for uneven bottoms, which consists in studying the motion of the free surface and the evolution of the velocity field of a layer of fluid under the following assumptions: the fluid is ideal, incompressible, irrotational, and under the only influence of gravity. Earlier works have set a good theoretical background for this problem. Its wellposedness has been discussed among others by Nalimov [START_REF] Nalimov | The Cauchy-Poison problem. (Russian) Dinamika Splošn. Sredy Vyp. 18 Dinamika Zidkost. so Svobod[END_REF], Yasihara [START_REF] Yosihara | Gravity waves on the free surface of an incompressible perfect fluid of finite depth[END_REF], Craig [START_REF] Craig | An existence theory for water waves and the Boussinesq and the Korteweg-de Vries scaling limits[END_REF], Wu [START_REF] Wu | Well-posedness in sobolev spaces of the full water wave problem in 2-D[END_REF], [START_REF] Wu | Well-posedness in sobolev spaces of the full water wave problem in 3-D[END_REF] and Lannes [START_REF] Lannes | Well-posedness of the water waves equations[END_REF]. Nevertheless, the solutions of these equations are very difficult to describe, because of the complexity of these equations, they are often replaced for practical purposes by approximate asymptotic systems. The most important examples are the Green-Naghdi equations (GN) which is a widely used model in coastal oceanography and the Camassa-Holm equation which is the unidirectional GN system and also Boussinesq, KdV equations ( [START_REF] Green | A derivation of equations for wave propagation in water of variable depth[END_REF][START_REF] Israwi | Variable depth KDV equations and generalizations to more nonlinear regimes[END_REF][START_REF] Israwi | Large time existence for 1D Green-Naghdi equations[END_REF]), the Shallow-Water equations, Camassa-Holm model, the Boussinesq systems, and KdV equation; their range of validity depends on the physical characteristics of the flow under consideration. The main goal of this article is to derive in a highly nonlinear regime with topographical effects a new linear class of asymptotic models of the Camassa-Holm and KdV equations. These new linear models have the same accuracy as the standard equations and can be solved explicitly. In order to validate our approximation, we give the explicit solutions of the linear asymptotic models and we illustrate their accuracy through an example by computing the residue in several norms that we obtain when we plug the explicit solutions into the standard equations.We also prove numerically the property of breaking wave for the Camassa-Holm equation with non flat bottom. ) is defined on Ω t = {(x, z), -1+βb (α) (x) < z < εζ(t, x)} (i.e. the velocity field is given by v = ∇ x,z ϕ):

(1)

                 µ∂ 2 x ϕ + ∂ 2 z ϕ = 0, at -1 + βb (α) < z < εζ, ∂ z ϕ -µβα(∂ x b) (α) ∂ x ϕ = 0 at z = -1 + βb (α) , ∂ t ζ - 1 µ (-µε∂ x ζ∂ x ϕ + ∂ z ϕ) = 0, at z = εζ, ∂ t ϕ + 1 2 (ε(∂ x ϕ) 2 + ε µ (∂ z ϕ) 2 ) + ζ = 0 at z = εζ.
The dimensionless parameters are defined as :

ε = a h 0 , µ = h 2 0 λ 2 , β = b 0 h 0 ;
where a is a typical amplitude of the waves; λ is the wavelength, b 0 is the order of amplitude of the variations of the bottom topography; λ/α is the wavelength of the bottom variations; h 0 is the reference depth. We also recall that b

(α) (x) = b(αx).
The parameter ε is often called nonlinearity parameter; while µ is the shallowness parameter. Asymptotic models from (1) are derived by making assumptions on the size of ε, β, α, and µ. In the shallow-water scaling (µ 1), one can derive (ε, β and α do not need to be small) the so-called Green-Naghdi equations (see [START_REF] Green | A derivation of equations for wave propagation in water of variable depth[END_REF][START_REF] Lannes | Derivation of asymptotic two-dimensional time-dependent equations for surface water wave propagation[END_REF] for a derivation and [START_REF] Alvarez-Samaniego | Large time existence for 3D water-waves and asymptotics[END_REF][START_REF] Israwi | Large time existence for 1D Green-Naghdi equations[END_REF] for a rigorous justification). For one-dimensional surfaces and over uneven bottoms these equations couple the free surface elevation ζ to the vertically averaged horizontal component of the velocity, ( 2)

u(t, x) = 1 1 + εζ -βb (α) εζ -1+βb (α) ∂ x ϕ(t, x, z)dz
and can be written as:

(3)

         ∂ t ζ + ∂ x (hu) = 0, (1 + µ h T [h, βb (α) ])[∂ t u + εu∂ x u + ∂ x ζ + µε - 1 3h ∂ x (h 3 (u∂ 2 x u) -(∂ x u) 2 ) + [h, βb (α) ]u = 0 where h = 1 + εζ -βb (α)
and

T [h, βb (α) ]W = - 1 3 ∂ x (h 3 ∂ x W ) + β 2 ∂ x (h 2 (∂ x b) (α) )W + β 2 h((∂ x b) (α) ) 2 W,
while the purely topographical term [h, βb (α) ]u is defined as:

[h, βb (α) ]u = β 2h [∂ x (h 2 (u∂ x ) 2 b (α) ) -h 2 ((u∂ 2 x u) -(∂ x u) 2 )(∂ x b) (α) ] +β 2 ((u∂ x ) 2 b (α) )(∂ x b) (α) .
We remark that the Green-Naghdi equations can then be simplified over uneven bottoms into (4)

ζ t + [hu] x = 0 u t + ζ x + εuu x = µ 3h [h 3 (u xt + εuu xx -εu 2 x )] x ,
where O(µ 2 ) terms have been discarded, and provided that the parameters satisfy θ = (α, β, ε, µ) ∈ ℘, where the set ℘ is defined as

℘ = {(α, β, ε, µ) such that ε = O( √ µ), βα = O(µ), βα = O(ε), (5)
βα 3/2 = O(µ 2 ), βαε = O(µ 2 )}.
In order to obtain the KdV equation (called KdV-top) originally derived in [START_REF] Kirby | Nonlinear Ocean surface waves[END_REF][START_REF] Svendsen | A direct derivation of the KDV equation for waves on a beach, and discussion of it's implications[END_REF][START_REF] Dingemans | Water waves propogation over uneven bottoms[END_REF], stronger assumptions on ε, β, α and µ must be made namely that the paramters belong to the subset ℘ ⊂ ℘ defined as:

(6) ℘ = {(α, β, ε, µ) such that ε = O(µ), αβ = O(ε), α 3/2 β = O(ε 2 )}.
Neglecting the O(µ 2 ) terms, one obtains from (4) the following Boussinesq system: (7)

ζ t + [hu] x = 0 u t + ζ x + εuu x = µ 3 (c 4 u xt ) x ,
where c = 1 -βb (α) .

Explicit solutions for the Kdv-top equations

In this section, attention is given to the weakly non linear regime ε = O(µ). We investigate several situations satisfying the condition (6) on the parameters ε, β, α and µ.

2.1. The continuous case.

ζ t + Γ 1 ζ + 3 2c εζζ x + 1 6 µc 5 ζ xxx = 0,
and

Γ 1 ζ = 1 2 (c ζ x + ∂ x (c ζ)),
where c = 1 -βb (α) . We assume that ( 6) is satisfied without any further assumptions; to this regime corresponds the so-called KdV-top (or original) model [START_REF] Craig | An existence theory for water waves and the Boussinesq and the Korteweg-de Vries scaling limits[END_REF]. It is related to the Boussinesq equations in the meaning of consistency (see below) and it was originally derived in [START_REF] Kirby | Nonlinear Ocean surface waves[END_REF][START_REF] Svendsen | A direct derivation of the KDV equation for waves on a beach, and discussion of it's implications[END_REF][START_REF] Dingemans | Water waves propogation over uneven bottoms[END_REF]. We list here some of the properties of this model. The proof of all the results below can be found in [START_REF] Israwi | Variable depth KDV equations and generalizations to more nonlinear regimes[END_REF]. Let us first define two different kinds of consistency, namely, L ∞ and H s consistency.

Definition 1. Let ℘ 0 ⊂ ℘ be a family of parameters (with ℘ as in ( 5)). A family

(ζ θ , u θ ) θ∈℘0 is L ∞ -consistent (or is an L ∞ -explicit solution with correctors of order O(µ 2 )) on [0, T ε ]
with the GN equations ( 4), if for all θ ∈ ℘ 0 (and denoting

h θ = 1 + εζ θ -βb (α) ),    ζ θ t + [h θ u θ ] x = µ 2 r θ 1 u θ t + ζ θ x + εu θ u θ x = µ 3h θ [(h θ ) 3 (u θ xt + εu θ u θ xx -ε(u θ x ) 2 )] x + µ 2 r θ 2 with (r θ 1 , r θ 2 ) θ∈℘0 bounded in (L ∞ ([0, T ε ] × R)) 2 .
When the residual is bounded in H s and not in L ∞ , we talk about H s -consistency. When s > 1/2, this H s -consistency is obviously stronger then the L ∞ -consistency. Definition 2. Let ℘ 0 ⊂ ℘ be a family of parameters (with ℘ as in ( 5)). A family (ζ θ , u θ ) θ∈℘0 is H s -consistent of order s ≥ 0 (or is an H s -explicit solution with correctors of order O(µ 2 )) on [0, T ε ] with the GN equations ( 4), if for all θ ∈ ℘ 0 , (and denoting h θ = 1 + εζ θ -βb (α) ),

ζ θ t + [h θ u] x = µ 2 r θ 1 u θ t + ζ θ x + εu θ u θ x = µ 3 1 h θ [(h θ ) 3 (u θ xt + εu θ u θ xx -ε(u θ x ) 2 )] x + µ 2 r θ 2 with (r θ 1 , r θ 2 ) θ∈℘0 bounded in (L ∞ ([0, T ε ], H s (R))) 2 . Remark 1.
The definitions can be adapted to define L ∞ and H s consistency with the Boussinesq equations ( 7) rather then the GN equations (4). We denote by

H ∞ (R) = ∩ s H s (R).
For the KdV-top model [START_REF] Craig | An existence theory for water waves and the Boussinesq and the Korteweg-de Vries scaling limits[END_REF], H s -consistency cannot be established, but L ∞consistency holds as shown below (see [START_REF] Israwi | Variable depth KDV equations and generalizations to more nonlinear regimes[END_REF]):

Theorem 1. Let s > 3 2 , b ∈ H ∞ (R) and ζ 0 ∈ H s+1 (R). For all θ ∈ ℘ ℘ = {(α, β, ε, µ) such that ε = O(µ), αβ = O(ε), α 3 2 β = O(ε 2 )},
we obtain the following properties :

• there exists T > 0 and a unique family of solutions

(ζ θ ) θ∈℘ to (8) bounded in C([0, T ε ]; H s+1 (R)) with initial condition ζ 0 ;
• the family (ζ θ , u θ ) θ∈℘ with (omitting the index θ)

(9) u := 1 c ζ - 1 2 x -∞ c x c ζ ds - ε 4c 2 ζ 2 + µ 1 6 c 4 ζ xx is L ∞ -consistent on [0, T ε ]
with the equations ( 7). Remark 2. The term x -∞ cx c ζ ds does not necessarily decay at infinity, and this the reason why H s -consistency does not hold in general. The problem of the convergence of the solution of ( 8) to the solution of ( 7) remains open; 2.1.2. The gentle model. In a first stage, we restrict here our attention to parameters ε, β, α and µ such that

(10) ε = O(µ), β = O(ε), α = O(ε).
These conditions are stronger than (α, β, ε, µ) ∈ ℘ ; we remark in particular that under the condition [START_REF] Dingemans | Water waves propogation over uneven bottoms[END_REF], the model ( 8) can be written after neglecting the O(µ 2 ) terms as :

(11) ζ t + cζ x + 3 2 εζζ x + 1 6 µζ xxx = 0,
This model [START_REF] Green | A derivation of equations for wave propagation in water of variable depth[END_REF] will be called gentle model since it is able to handle H s -consistent on [0, T ε ] with the equations ( 7), and a full justification (convergence) is established for this model (see [START_REF] Israwi | Variable depth KDV equations and generalizations to more nonlinear regimes[END_REF]).

2.1.3. The strong model. We consider here stronger variations of the values of α, i.e. :

(12) ε = O(µ), β = O(ε), α = O(ε 2/3 ).
The original equation becomes

(13) ζ t + Γ 1 ζ + 3 2 εζζ x + 1 6 µζ xxx = 0,
Remark 3. Note that the model 13 is L ∞ -consistent with 7, for more details, see theorem 1

2.2. The Explicit solutions. In this section we show that there is an explicit expression family ζ θ H s -consistent with the original model KdV-top 8 for some θ = (α, β, ε, µ).

Theorem 2. Let b ∈ H ∞ (R), ε = O(µ), β = α = O(ε), ζ f solution of the system (
(14) ∂ t ζ b + ∂ x ζ b = 1 2 b (α) ∂ x ζ f , then ζ = ζ f + βζ b is H s -consistent
with the gentle KdV model [START_REF] Green | A derivation of equations for wave propagation in water of variable depth[END_REF].

Proof. For the sake of simplicity, we denote by O(µ) any family of functions (f µ ) 0<µ<1

such that 1 µ f µ 0<µ<1 remains bounded in L ∞ ([0, T ε ], H r (R))
, for possibly different values of r. The same notation is also used for real numbers, e.g ε = O(µ), but this should not yield any confusion. We seek ζ b such that if

ζ = ζ f + βζ b ,
and ζ f solves (8) with flat bottom (c = 1) then the equation ( 11) is satisfied up to O(µ 2 ). Indeed, we use the fact that

c = 1 - 1 2 βb (α) ∂ x ζ f + O(β 2 ), β = O(µ)
and : ( we denote by

ζ f s = ∂ s ζ f .) (15)
ζ f t + ζ f x + 3 2 εζ f ζ f x + 1 6 µζ f xxx = 0,
one can chosse ζ b as solution to the following linear transport equation

β∂ t ζ b + β∂ x ζ b = 1 2 βb (α) ∂ x ζ f ,
and the conditions given in the statement of the theorem on ζ b yield the result

The next result is concerning the strong KdV equation [START_REF] Van Groesen | Uni-directional waves over slowly varying bottom. I. Derivation of a KdV-type of equation[END_REF].

Theorem 3. Let b ∈ H ∞ (R), ε = O(µ), β = O(ε), α = O(ε 2/3 ) ζ f solution of the system (
∂ t ζ b + ∂ x ζ b = 1 2 b (α) ∂ x ζ f + 1 4 α(b x ) (α) ζ f , then ζ = ζ f + βζ b is H s -consistent (16)
with the strong KdV model [START_REF] Van Groesen | Uni-directional waves over slowly varying bottom. I. Derivation of a KdV-type of equation[END_REF].

Proof. One can adapt the proof of this Theorem in the same way of the proof of Theorem 2, the only extra term here is

1 2 c x ζ which it is not of order O(µ 2 ) but 1 2 c x ζ = - 1 4 βα(b x ) (α) ζ f + O(β 2 ),
and we omit the proof here.

In this section of the paper we will prove that there exists an explicit solution for the KdV equations ( 11) and ( 13) with correctors of order O(µ 2 ).

2.3.

Analytic solution for the model [START_REF] Holden | [END_REF]. In the case of flat bottoms, analytical solutions are well-known for the model [START_REF] Holden | [END_REF], and consist of solitary-waves. Let us consider the following function parametrized by c 1

Ψ(x) = 2 c 1 sech 2 3c 1 ε 2µ x ,
therefore, the analytical solution for the flat bottom of ( 8): (the index f stands for "flat bottom") is equal to

(17) ζ f (t, x) = Ψ(x -c t),
with a real velocity c c = 1 + εc 1 .

2.4.

Explicit solution for the models [START_REF] Green | A derivation of equations for wave propagation in water of variable depth[END_REF] and [START_REF] Van Groesen | Uni-directional waves over slowly varying bottom. I. Derivation of a KdV-type of equation[END_REF]. In this subsection we prove the main result of this article, which shows that the KdV-top equations [START_REF] Green | A derivation of equations for wave propagation in water of variable depth[END_REF][START_REF] Van Groesen | Uni-directional waves over slowly varying bottom. I. Derivation of a KdV-type of equation[END_REF] has an explicit solution with correctors of order O(µ 2 ).

Theorem 4. Let b ∈ H ∞ (R), ε = O(µ), β = O(µ), ζ f
given by the expression [START_REF] Coclite | A Convergent finite difference scheme for the Camassa-Holm equation with general H 1 inital data[END_REF] and

g(t, x) = 1 2 b (α) (x)∂ x ζ f (t, x). Let also the initial condition ζ 0 (x) = ζ f (0, x) + βζ 0 b (x), where ζ 0 b is given in H s (R). Then, ζ with (18) ζ = ζ f + β ζ 0 b (x -t) + t 0 g(s, x -t + s) ds ,
is an H s -explicit solution with correctors of order O(µ 2 ) on [0, T ε ] for the gentle KdV equation [START_REF] Green | A derivation of equations for wave propagation in water of variable depth[END_REF].

Remark 4. In a similar way we can obtain a same formula [START_REF] Iguchi | A long wave approximation for capillary-gravity waves and the Kawahara equation[END_REF] for [START_REF] Van Groesen | Uni-directional waves over slowly varying bottom. I. Derivation of a KdV-type of equation[END_REF].

Proof. We know from Theorem 2 that

ζ = ζ f + βζ b ,
is H s -consistent with the KdV-top equation [START_REF] Green | A derivation of equations for wave propagation in water of variable depth[END_REF], where,

ζ b = ζ 0 b (x -t) + t 0 g(s, x -t + s) ds,
since it is solution of the following linear equation :

∂ t ζ b + ∂ x ζ b = g(t,
x), therefore one get easily the result.

Explicit solutions for the Camassa-Holm-like equations

Now we consider the generalizations to more nonlinear regimes of the KdV-top equation derived in [START_REF] Constantin | The hydrodynamical relevance of the Camassa-Holm and Degasperis-Processi equations[END_REF] for flat bottoms and [START_REF] Israwi | Variable depth KDV equations and generalizations to more nonlinear regimes[END_REF] for variable bottoms.

3.1. The continuous case. [START_REF] Israwi | Variable depth KDV equations and generalizations to more nonlinear regimes[END_REF]) :

The original model. The family of equations on the surface elevation ζ (see

ζ t + cζ x + 1 2 c x ζ + 3 2c εζζ x - 3 8c 3 ε 2 ζ 2 ζ x + 3 16c 5 ε 3 ζ 3 ζ x +µ( Ãζ xxx + Bζ xxt ) = εµ Ẽζζ xxx + εµ ∂ x ( F 2 ζ)ζ xx + ζ x ∂ 2 x ( F 2 ζ) , (19)
where

à = Ac 5 -Bc 5 + Bc Ẽ = Ec 3 - 3 2 Bc 3 + 3 2c B F = F c 3 - 9 2 Bc 3 + 9 2c B,
(with some conditions on A, B, E, and F ), called here original model, can be used to construct an approximate solution consistent with the Green-Naghdi equations.

Theorem 5. Let s > 3 2 , q ∈ R, b ∈ H ∞ (R) and ζ 0 ∈ H s+1 (R). Assume that A = q, B = q - 1 6 E = - 3 2 q - 1 6 , F = - 9 2 q - 5 24 .
For all θ ∈ ℘ such that

℘ = {(α, β, ε, µ) such that ε = O( √ µ), βα = O(µ), βα = O(ε), βα 3/2 = O(µ 2 ), βαε = O(µ 2 )}
we obtain

• there exists T > 0 and a unique family of solutions

(ζ θ ) θ∈℘ to (19) bounded in C([0, T ε ]; H s+1 (R)) with initial condition ζ 0 ; • the family (ζ θ , u θ ) θ∈℘ with (omitting the index θ) u := 1 c ζ + c 2 c 2 + εζ - 1 2 x -∞ c x c ζ - ε 4c 2 ζ 2 - ε 2 8c 4 ζ 3 + 3ε 3 64c 6 ζ 4 (20) -µ 1 6 c 3 ζ xt + εµc 2 1 6 ζζ xx + 1 48 ζ 2 x is L ∞ -consistent on [0, T ε ]
with the GN equations ( 4). Remark 5. If we take q = 1 12 , b = 0, i.e if we consider a flat bottom, then one can recover the equation ( 19) of [START_REF] Constantin | The hydrodynamical relevance of the Camassa-Holm and Degasperis-Processi equations[END_REF]:

ζ t + ζ x + 3 2 εζζ x - 3 8 ε 2 ζ 2 ζ x + 3 16 ε 3 ζ 3 ζ x + µ 12 (ζ xxx -ζ xxt ) = - 7
ζ t + cζ x + 3 2 εζζ x - 3 8 ε 2 ζ 2 ζ x + 3 16 ε 3 ζ 3 ζ x + µ 12 (ζ xxx -ζ xxt ) = - 7 24 εµ(ζζ xxx + 2ζ x ζ xx ). ( 22
)
This model ( 22) is called gentle model since it is only able to handle gentle variations of bottom topography.

3.1.3. The strong model. We consider here stronger variations of the parameters, i.e. :

(23) q = 1 12 ε = √ µ, β = µ 3/2 , α = ε 2/3 . ζ t + Γ 1 ζ x + 3 2 εζζ x + - 3 8 ε 2 ζ 2 ζ x + 3 16 ε 3 ζ 3 ζ x + µ 12 (ζ xxx -ζ xxt ) = - 7 24 εµ(ζζ xxx + 2ζ x ζ xx ). ( 24
)
This model ( 24) is called strong model since it is able to handle strong variations of wavelength of the bottom . One can remark that we can easily prove for these CH-Like-models the same results proved in the previous section for the KdV-topmodels: Theorem 6. Let b ∈ H ∞ (R), the parameters verify the conditions of the gentle model and ζ f solution of the system [START_REF] Johnson | A modern introduction to the mathematical theory of water waves[END_REF] and ζ b solution of the following linear equation:

(25) ∂ t ζ b + ∂ x ζ b = 1 2 b (α) ∂ x ζ f , then ζ = ζ f + βζ b is H s -consistent
∂ t ζ b + ∂ x ζ b = 1 2 b (α) ∂ x ζ f + 1 4 α(b x ) (α) ζ f , then ζ = ζ f + βζ b is H s -consistent with the strong CH model (24). (26)

Numerical validation

4.1. Linear case of b α . In this section we present several numerical examples in order to justify our theoretical results. Firstly, the gentle model ( 11) is considered. We distinguish two cases : b α linear and b α non linear. In case of linear case , we compute explicitly ζ(t, x) given by [START_REF] Lannes | Well-posedness of the water waves equations[END_REF], then we substitute this explicit solution in the gentle model and we compute the residue. We start computing explicitly [START_REF] Iguchi | A long wave approximation for capillary-gravity waves and the Kawahara equation[END_REF]. To simplify, we note V = 3c 1 ε 2µ Thus , we get

∂ x ζ f = -4c 1 V tanh(V (x -c t))sech 2 (V (x -c t)) Therefore g(t, x) = -2αc 1 xV tanh(V (x -c t))sech 2 (V (x -c t)) hence ( 27
) t 0 g(s, x -t + s)ds = -2αc 1 V t 0 (x -t + s) tanh(V (x -t + s(1 -c )))× × sech 2 (V (x -t + s(1 -c )))ds Let I(s) = ((x -t + s) tanh(V (x -t + s(1 -c )))sech 2 (V (x -t + s(1 -c ))))ds.
Set u = x -t + s then we get

I(s) = u tanh(V (u -c (u -x + t)))sech 2 (V (u -c (u -x + t)))du = u tanh(V ((1 -c )u + c (x -t)))sech 2 (V ((1 -c )u + c (x -t)))du Now consider w = V ((1 -c )u + (x -t)c
), hence , integration by part leads

I(s) = 1 V 2 (1 -c ) 2 w tanh(w)sech 2 wdw - (x -t)c V V 2 (1 -c ) 2 tanh(w)sech 2 (w)dw = 1 V 2 (1 -c ) 2 - 1 2 wsech 2 (w) + 1 2 sech 2 (w)dw + 1 2 (x -t)c V V 2 (1 -c ) 2 sech 2 (w) = - 1 2V 2 (1 -c ) 2 wsech 2 (w) + 1 2V 2 (1 -c ) 2 tanh(w) + 1 2 (x -t)c V V 2 (1 -c ) 2 sech 2 (w)
Taking into consideration that tanh(w) = 1 2 sinh(2w)sech 2 (w) and w = V (x -t +

(1 -c )s) we get

I(s) = 1 4(c -1) 2 V 2 [(2(c -1)V (s -t + x) + sinh(2V (x -t + (1 -c )s)))] sech 2 (V (x-t+(1-c )s)).

By definition

t 0 g(s, x -t + s)ds = -2c 1 αV (I(t) -I(0)) so t 0 g(s, x -t + s)ds = -2c 1 αV 4(c -1) 2 V 2 2(c -1)V x + sinh(2V (x -c t)) × (28) ×sech 2 (V (x -c t)) -2(c -1)V (x -t) + (29) sinh(2V (x -t)) sech 2 (V (x -t))
And hence the solution is given by the following formula :

(30) ζ = ζ f + β[ζ 0 b (x -t) + t 0 g(s, x -t + s)ds]
We define the following residues :

R ∞ (µ) = ζ t + cζ x + 3 2 εζζ x + 1 6 µζ xxx ∞ R 0 (µ) = ζ t + cζ x + 3 2 εζζ x + 1 6 µζ xxx L 2 R 1 (µ) = ζ t + cζ x + 3 2 εζζ x + 1 6 µζ xxx H 1
The results are illustrated in the Figure 1, where the red, violet, blue and green curves represent R 1 (µ), R 0 (µ), R ∞ (µ) and µ 2 respectively. We note that the asymptotic behaviour of the residues are of order O(µ 2 ) which justify our theoretical results. 16). the solution is given [START_REF] Schneider | C The long-wave limit for the water wave problem I. The case of zero surface tension[END_REF] With I(s) = (x -t + s)sech 2 (V (x -t + s -c s)) tanh(V (x -t + s -c s))ds and

with g(t, x) = 1 2 b α ∂ x ζ f + 1 4 αb α x ζ f For this we set ζ b 0 (x) = exp(-x 2 /10) ε = β = µ α = µ 2/3 In this case b α (x) = αx, so g(t, x) = -2αxc 1 V tanh(V (x -c t))sech 2 (V (x -c t)) + 1 2 α 2 c 1 sech 2 (V (x -c t))
J(s) = sech 2 (V (x -t + s -c s))ds I(s) is computed in the previous section. By taking u = V (x -t + (1 -c )s), we deduce J(s) = 1 V (1 -c ) sech 2 (u)du = 1 V (1 -c ) tanh(u) = 1 V (1 -c ) tanh(V (x -t + (1 -c )s))
Similarly the solution is given by the following formula

(31) ζ = ζ f + β[ζ 0 b (x -t) + t 0 g(s, x -t + s)ds]
We define the following residues :

R ∞ (µ) = ζ t + Γ 1 ζ + 3 2 εζζ x + 1 6 µζ xxx ∞ Figure 3. The Residues R 0 (µ), R ∞ (µ) and R 1 (µ) R 0 (µ) = ζ t + Γ 1 ζ + 3 2 εζζ x + 1 6 µζ xxx L 2 R 1 (µ) = ζ t + Γ 1 ζ + 3 2 εζζ x + 1 6 µζ xxx H 1
The results are illustrated in the Figure 2, where the red, violet, blue and green curves represent R 1 (µ), R 0 (µ), R ∞ (µ) and µ 2 respectively. 4.2. Nonlinear case of b α . In this section we take b α (x) = sin(αx), note that the integral in the formula [START_REF] Iguchi | A long wave approximation for capillary-gravity waves and the Kawahara equation[END_REF] for the models (11) and 1(3) is computed numerically by using the composite trapezoidal rule. The numerical results are illustrated in the figures 3 and 4 respectivly.

Wave breaking

We say that there is wave breaking for a scalar equation as Camass-Holm equations, if there exists a time 0 < t ,µ < ∞ and solutions f to this equation such that f ∈ L ∞ [0, t ,µ ] × R and lim t→t ,µ |∂ x f (t, .)| ∞ = ∞. Global well-posedness results for the KdV equation show that wave breaking cannot occur for this model. This observation led many others to look for other nonlinear dispersive models that could model breaking waves, as the Camassa-Holm-like equation [START_REF] Korteweg | On the change of form of long waves advancing in a rectangular canal and on a new type of long stationary waves[END_REF]. It indeed possess smooth solutions that develop singularities according to the wave breaking scenario described above: the solution remains bounded but its slope becomes unbounded. In this section we will prove numerically this property for the Camassa-Holm equation with non flat bottom [START_REF] Korteweg | On the change of form of long waves advancing in a rectangular canal and on a new type of long stationary waves[END_REF]. It's very hard to solve explicitly the equation [START_REF] Korteweg | On the change of form of long waves advancing in a rectangular canal and on a new type of long stationary waves[END_REF] for that reason we will construct an asymptotic explicit solution which is consistent with this equation. By using a linear model consistent The initial value ζ(0, x) = ζ f (0, x) + βζ b 0 (x) with ζ b 0 (x) = 3 sin(3x) is fixed . In the figure 7, 8 and 9, an approximation of ∂ x ζ(1, t) is plotted, and we vary the time. We note that there exist T * belongs the interval [40,45] such that the braking wave phenomena appears. In addition, when the value of µ decreases, the explosion time T * converges to a fixed value (approximately 44).

1. 2 .

 2 Boussinesq and Green-Naghdi equations. Parametrization the free surface by z = εζ(t, x) (with x ∈ R) and the bottom by z = -1 + βb (α) (x) (with Date: September 23, 2018.

Figure 1 .

 1 Figure 1. The Domain

2. 1 . 1 .

 11 The KdV-top (or original) model. The model studied in this section is the following (ζ is the elevation)[START_REF] Craig | An existence theory for water waves and the Boussinesq and the Korteweg-de Vries scaling limits[END_REF]

8 )

 8 with c = 1 and ζ b solution of the following linear equation:

  8) with c = 1 and ζ b solution of the following linear equation:

21 ) 3 . 1 . 2 .

 21312 24 εµ(ζζ xxx + 2ζ x ζ xx ). (The gentle model. Choosing q = 1 12 , ε = √ µ, α = ε and β = µ 3/2 the equation (19) reads after neglecting the O(µ 2 ) terms:

Theorem 7 .

 7 with the gentle CH model[START_REF] Johnson | On the development of a solitary wave moving over an uneven bottom[END_REF]. Let b ∈ H ∞ (R), the parameters verify the conditions of the strong model and ζ f solution of the system[START_REF] Johnson | A modern introduction to the mathematical theory of water waves[END_REF] and ζ b solution of the following linear equation:

4. 1 . 1 .

 11 Model 11. Firstly, the gentle model (11) is considered. For this we set b α (x) = αx, ζ b 0 (x) = exp(-x 2 /10) ε = β = α = µ

4. 1 . 2 .

 12 Model 13. In this section we consider ζ = ζ f + βζ b where ζ b is the solution of the linear equation (

Figure 2 .

 2 Figure 2. The Residues R 0 (µ), R ∞ (µ) and R 1 (µ)

Figure 4 .

 4 Figure 4. The Residues R 0 (µ), R ∞ (µ) and R 1 (µ)

Figure 5 .

 5 Figure 5. The Residues R 0 (µ), R ∞ (µ) and R 1 (µ)

Figure 7 . 1 Figure 8 .

 718 Figure 7. The Breaking wave phenomena for µ = 0.1

Finally we conclude

Graphically, we define the following residues :

We fix t = 1 and we vary µ. The results are illustrated in the Figure 5, where the green, red and blue curves represent µ 2 , R ∞ (µ), R 0 (µ) respectively.