^{1,2}

^{1,2}

^{3}

^{1}

^{2}

^{3}

We consider three inviscid, incompressible, irrotational fluids that are contained between the rigid walls

Since some early studies started in the 1950s (see [

However, to the best of our knowledge, the phenomenological models for more than two fluids mentioned at the beginning of this section have not been paralleled by any analytical study. This prompted us to develop a generalization of the NSP formulation to the case of three ideal fluids, separated by two free interfaces and limited above by a rigid lid. Namely, we consider three inviscid, incompressible, irrotational fluids that are confined between the rigid lids

Three fluids with two free internal interfaces.

The NSP formulation is given by the following equations.

In the next section, we derive the above equations, starting from the classic equations governing three ideal fluids separated by two free interfaces bounded above and below by rigid lids. Moreover, we derive conservation laws and integral identities for the three-fluid system for the NSP formulation.

Section

In Section

We recall the classic equations governing three ideal fluids separated by two free interfaces

We now obtain a weak formulation of (

We derive from (

Along the same lines used for the first layer, let us derive (

We define the basic functions

Similarly, let us consider the basic functions

For the top layer, we require that (

In [

Similarly, by setting to zero the coefficient of

Next, we obtain a last identity at third order, setting to zero the coefficient of

In order to derive weakly nonlinear equations, we nondimensionalize all physical variables in (

Then, (

The above assumptions imply that we are interested to derive asymptotic reductions of the NSP equations in the case of weakly nonlinear long waves.

We first expand (

We now turn our attention to the intermediate layer.

By equating the expansions of (

We now expand (

When we substitute (

The system of 3 generalized Boussinesq equations (

In order to obtain some interesting limiting equations, we now make the assumption of

In this section, we study a

Here below, we assume the following asymptotic expansions for

In terms of the new variables, one obtains

Considering the assumption of maximal balance together with (

We also assume unidirectional waves and

In order to remove secular terms, the right-hand side of (

Following the same steps as above for (

In the following, we will study only the

Finally, define the functions

The two coupled shallow water equations will be studied numerically as a function of the parameters entering the theory, in order to prove the existence of solitary waves and analyze their behaviour.

We now investigate whether (

To solve (

In general, we cannot find a solution to (

We use the above SPRZ scheme to solve for the modes

Figure

Amplitude versus negative speed for

Figure

After obtaining these results, it would now be of particular interest from the applied point of view to address the issue of a local breakdown of the nonlinear internal waves propagating in the density stratified fluid [

The authors declare that there is no conflict of interests regarding the publication of this paper.

The authors wish to thank M. J. Ablowitz for useful suggestions.