The purpose of this paper is to introduce and analyze a nonlinear, dispersive model for the one-way propagation of long internal waves of small amplitude along the interface of a two-layer system of fluids under gravity, surface tension and rotational effects. The model can be derived from different points of view; rotating- and nonrotating-fluid models proposed in the literature, mainly the Ostrovsky equation, [12, 25], and the Benjamin equation, [1, 3, 4, 5], respectively. The analysis of the new system, exposed in the present paper, is focused on mathematical aspects concerning well-posedness, conserved quantities and existence of solitary wave solutions. The main highlights are the following:
Sufficient conditions on the parameters of the model are given in order to obtain existence and uniqueness of solutions of the associated linear problem. The result makes use of the theory on oscillatory integrals and regularity of dispersive equations developed in [14] (see also [8, 15, 16, 19, 23]). The equation is shown to admit three conserved quantities by decaying to zero at infinity and smooth enough solutions. A Hamiltonian formulation is also derived. One of the relevant properties of nonlinear dispersive models for wave propagation is the existence of traveling-wave solutions of solitary type, [6, 7] (see [10, 13] and references therein for the case of internal waves). In this sense, and using the Concentration-Compactness theory, [22], the new model is proved to admit such solutions, under suitable conditions on the parameters. By using the Petviashvili’s iterative method, [28], to generate approximations to the solitary-wave profiles, several properties of the waves are analyzed by computational means. They concern the speed-amplitude relation, the asymptotic decay and the comparison with similar structures presented in classical rotating-fluid models like the Ostrovsky equation.
The paper is structured as follows. In Section 2, the model will be introduced, from the general problem of propagation of internal waves along the interface of a two-layer system and under the corresponding physical regime of validation. Its justification from existing rotating- and nonrotating-fluid models by incorporating new physical assumptions will be discussed. Section 2 is finished off with the analysis of linear well-posedness of the corresponding initial-value problem (IVP) and the derivation of functionals preserved by smooth enough solutions vanishing suitably at infinity. In particular, a Hamiltonian structure of the problem comes out from one of these quantities. Section 3 is focused on the existence of solitary-wave solutions. As a first approach we make a computational study, with a description of the numerical technique used to generate approximate solitary-wave profiles and the numerical illustration of some of their properties. Then a theoretical result of existence of these solutions, under suitable conditions on the parameters of the model, is established. These conditions for the existence will also help us to compare, by computational means, the proposed model with classical rotating-fluid models such as the Ostrovsky equation, with the aim of investigating the influence of the new physical properties assumed. Conclusions and future lines of research will be outlined in Section 4.
The following notation will be used throughout the paper. By
The two-layer interface problem for internal wave propagation, of interest for the present paper, is idealized in Figure 1. This consists of two inviscid, homogeneous, incompressible fluids of depths
Idealized model of internal wave propagation in a two-layer interface. Fig. 1
From this idealized system and in order to limit the physical regime of validation of the proposed model, some hypotheses are assumed. The first one is described in terms of the dimensionless parameters as
referred to the upper fluid layer and where
Amplitude (Fig. 2
Finally, capillary and gravity forces are assumed to be nonnegligible, as well as a dispersion effect due to the rotations of the fluids. These assumptions are translated to the following partial differential equation (PDE) for the evolution of the deviation of the interface
where
being one of the nonlocal terms of (1), then the general dispersive effects (surface tension and gravity) are controlled by the parameters
The equation (1) includes some well-known limiting cases from which its derivation can be justified. These can be rotating or nonrotating models. In the first case, one may start from the Ostrovsky equation, [2, 10, 11, 12, 13, 25, 26, 29, 30]
and include the hypothesis of a much larger density of the lower fluid,
and incorporate nonnegligible surface tension effects through the term associated to
if we assume (as in the case of the Ostrovsky equation with respect to the Korteweg-de Vries (KdV) equation, see [12]) that the rotational effects in the fluids are relevant enough to be included as a second nonlocal dispersive term
or equivalently
where
denotes the Fourier transform of
Due to the relation with the Benjamin equation (5), equation (1) will be sometimes referred as the rotation-modified Benjamin (RMBenjamin) equation.
We finally observe that, as in the case of other models, [18], an extension of (1) is obtained by considering general homogeneous nonlinearities
in such a way that (1) can be generalized to
The main theoretical results below will be established for (8), although the particular case of (1) (for which
This section concerns the well-posedness of the IVP of the linearized equation associated to (8)
Using the Fourier representation of (2)
then the application of the Fourier transform (in
where
The inversion of (10) allows to write formally the solution of (9) in the operational form
The following lemma will be used to estimate (11).
Let
while if
These two inequalities, under the hypotheses on
Following similar arguments to those of [20, 21, 32] for the case of the Ostrovsky (3) and the RMBO equations, (3) and (4) respectively, we have the following result on well-posedness of (9).
Recall that a change of variable allows to assume
for some constant
with Ω = (−∞, 0) ∪ (0, ∞). Now, using Lemma 1, observe that (cf. [20, 21])
and (13) follows.□
A second mathematical property is concerned with the existence of invariant quantites of (8) for smooth enough solutions. Similar results to those of other rotating models, like the Ostrovsky equation, or nonrotating fluid models like the Benjamin equation, can be derived in this case. The proof is straightforward.
A third mathematical property of (8), under study in the present paper, is concerned with the existence of solitary wave solutions. These are solutions of permanent form
where
The numerical approximation to (16) may give us a first approach about the existence of solitary wave solutions and some of their properties. In this section this will be illustrated for the case of (1), that is when
which may be solved iteratively. Among other alternatives presented in the literature, (see e. g. the review and references in [33]), the numerical resolution of (17) will be here performed with the Petviahsvili’s method, [28]. This is formulated as follows. Given an initial profile
where ℒ, 𝒩 are, respectively, the linear and nonlinear operators defined in (17) and 〈 ⋅, ⋅, 〉 denotes the Euclidean inner product. The application of the Petviashvili’s method requires, among other conditions, a nonlinearity 𝒩 of homogeneous type. Its degree of homogeneity determines the exponent of
which is obviously an approximation to (17) with approximate operators ℒ
where in (21),
with
The purpose of the implementation of (22), (23) is two-fold: having more certainty about the existence of solitary waves and deriving a computational way to obtain approximate profiles from which some properties of the waves and their dynamics can be discussed. Thus, Figure 3 shows some computed profiles corresponding to different values of the speed and for
Numerical approximation with Fig. 3
These two properties are confirmed by the following figures. Figure 4 displays the phase portraits of the profiles computed in Figure 3 and the oscillatory decay is clearly observed. By fitting the values close to the origin, the results suggest that the waves decay algebraically, as in the cases of the Benjamin equation, [1] and the Ostrovsky equation, [9, 10].
Numerical approximation with Fig. 4
Figure 5 shows the behaviour of the maximum positive excursion
Numerical approximation with Fig. 5
Two-pulse for Fig. 6
These computations motivate to study the existence of solitary wave solutions of (8) theoretically. This is developed in the following subsection.
We consider the space
(in [23] this is denoted by
or, equivalently
(see (30)), where
The purpose in this section is to discuss the existence of solitary-wave solutions of (8) in terms of the parameters
We define the functionals
and consider, for
In order to prove that
The proof of Lemma 10 is based on the following estimates of
In the case of (i): In the case of (ii): for some In the case of (iii): for some In the case of (iv): for some
The proof of (31)-(34) is as follows.
The proof of (31) is trivial since For the proof of (32) we write, [23] Then, for any This is applied to and choosing For the proof of (33), we use (29) and similar arguments to those of the previous proof to choose Proof of (34): In this case, the same strategy as above is applied twice. First, we have Now the two coefficients are positive when Note that (36) implies that we need 4
Once (31)-(34) is proved, we can use (28), (29) and the estimate (14) of [8] to have
for some constant
for some constant
which implies
for any
□
Two additional properties will be used to prove the existence result:
From (30) (see also formula (17) in [8], along with (29) of the present paper) we obtain the coercivity of for any for all
The main result of existence is the following.
From coercivity of
then
and normalizing (by taking
If we apply the Concentration-Compactness Lemma, [22], to
Compactness: there exist Vanishing: For every Dichotomy: there exists for
The next step is ruling out possibilities (b) and (c). Here the arguments are similar to those of, for example, [17]. Assume that (b) holds. Using (37) and the homogeneity of
for all
for
that is
Assume now that (c) holds. Define cutoff functions
Then
Since
where
Taking And, finally, if which is a contradiction. Then As above, if And, again, if
So (c) is also ruled out and therefore compactness (a) holds. Now we prove that (a) implies the existence of a minimizer. (Again the arguments are similar to, e. g. [8, 17].) Since
Now we prove that
By compactness of
Therefore, if
for
for
which implies that
Therefore
Fig. 7
Fig. 8
The purpose of this section is to compare, by computational means and through the corresponding solitary waves, the equation (1) with the Ostrovsky equation, the classical model for internal waves in rotating fluids, which is the limiting case of (1) by taking
A first observation in this sense is concerned with the behaviour of the amplitude of the solitary waves of (1) as function of
Amplitude vs Fig. 9
We also consder the behaviour of the limiting value of the speed to ensure the existence of solitary waves as function of
Fig. 10
These two observations can explain the comparisons between the solitary waves of (1) and of the Ostrovsky equation shown in Figures 11 and 12. Figure 11 depicts the profiles corresponding to each equation for two values of the speed. According to this and the previous figures, note that the presence of the nonlocal term in (1) with
RMBenjamin vs Ostrovsky equations. Computed solitary wave profiles with (a) Fig. 11
RMBenjamin vs Ostrovsky equations. Speed-amplitude relations.Fig. 12
This is confirmed when we compare the behaviour of the maximum (
The present paper introduces a nonlinear dispersive nonlocal model for the propagation of internal waves in a two-layer system and under the presence of gravity, surface tension and rotational forces. The model can be derived from the inclusion of gravity effects in the rotating fluid model given by the Ostrovsky equation or by incorporating a dispersive, rotational component in the nonrotating model of the Benjamin equation. The proposed system can also be generalized by including nonlinear terms from quadratic to any of homogeneous type with degree of homogeneity greater than two.
Three mathematical aspects of the model and its generalizations are analyzed. The first one is concerned with linear well-posedness and here sufficient conditions for existence and uniqueness of solution of the corresponsing IVP of the linear problem are established by using the theory developed in [14] and in terms of the parameters (with the corresponding physical meaning) of the equation. The second result is the derivation of three conservations laws and the Hamiltonian formulation, in accordance with its limiting cases of the Benjamin and Ostrovsky equations. Finally, the existence of solitary wave solutions is discussed, computationally and analytically. The generation of approximate solitary-wave profiles, described and developed in the present paper, gives a first indication of existence of solitary waves, suggests some of their properties (such as the amplitude-speed relation and the oscillatory decay) and allows to make comparisons between the proposed model and the classical rotating-fluid model given by the Ostrovsky equation. On the other hand, a theoretical result of existence, in terms of the parameters of the equation, is derived by using the Concentration-Compactness theory, [22], as in related rotating and nonrotating models.
Some open questions for a future research can be finally mentioned:
The first one is to make progress in the study of linear and nonlinear well-posedness, as well as in the proof of regularity and asymptotic decay of the solitary wave solutions, suggested by the numerical experiments. Another research line is in the study of the stability of the solitary waves, both orbital and asymptotic, either theoretically or computationally. In this last point, the use of efficient numerical integrators, in order to have accurate long term simulations, is required. A third open question is the analysis of the influence of the rotational effects from nonrotating-fluid models in more detail; in particular, it would be worth studying the weak rotation limit to the Benjamin equation, in a sort of comparison witrh the analogous property between the Ostrovsky equation and its weak rotation limit model, the KdV equation, [19, 23, 31].