Generalized Solitary Waves in the Gravity-Capillary Whitham Equation
We study the existence of traveling wave solutions to a unidirectional shallow water model which incorporates the full linear dispersion relation for both gravitational and capillary restoring forces. Using functional analytic techniques, we show that for small surface tension (corresponding to Bond numbers between and ) there exists small amplitude solitary waves that decay to asymptotically small periodic waves at spatial infinity. The size of the oscillations in the far field are shown to be small beyond all algebraic orders in the amplitude of the wave. We also present numerical evidence, based on the recent analytical work of Hur & Johnson, that the asymptotic end states are modulationally stable for all Bond numbers between and .
1.1. “Full dispersion” models
It is well known that the Korteweg-de Vries (KdV) equation
approximates the full water wave problem in the small amplitude, long wavelength regime [26, Section 7.4.5]   . Here, corresponds to the fluid height at spatial position at time , corresponds to the undisturbed depth of the fluid, and is the acceleration due to gravity. At least in part, the agreement in this asymptotic regime can be understood by noting that the phase speed of the water wave problem expands for as
so that the KdV phase speed agrees to second order in with that of the full water wave problem.
The KdV equation admits both solitary and periodic traveling wave solutions which are
nonlinearly stable in appropriate senses   and these results have pointed the way towards (at least some) similar results for the full water wave problem
Naturally, however, the KdV phase speed is a terrible approximation of for
even moderate frequencies. It should come as no surprise then that KdV fails to exhibit many high-frequency phenomena
The above observations led Whitham  to state “It is intriguing to know what kind of simpler mathematical equations (than the physical problem) could include [peaking and breaking].” In response to his own question, Whitham put forward the model
where here is a Fourier multiplier operator on defined via
see [39, p. 477] By construction, the above pseudodifferential equation, modernly referred to as the “Whitham equation” or “full dispersion equation”, has a phase speed that agrees exactly with that of the full water wave problem. Since (2) balances both the full water wave dispersion with a canonical shallow water nonlinearity, Whitham conjectured that the equation (2) would be capable of predicting both breaking and peaking of waves.
And in fact it does. The Whitham equation (2) has recently been shown
1.2. Including surface tension
It is thus natural to consider the existence and behavior of solutions when additional physical effects are included. In this paper, we incorporate surface tension and consider the following pseudodifferential equation
Here, , , and are as in (2) above, and is a Fourier multiplier operator on with symbol
This symbol gives exactly the phase speed for the full gravity-capillary wave problem in the irrotational setting [25, 39]. The parameter is the coefficient of surface tension, while both and are as in (2). The properties of the symbol above depend on the non-dimensional ratio
which is referred to as the Bond number. When , corresponding to “strong” surface tension, the phase speed is monotone increasing for with high-frequency asymptotics for , while for “weak” surface tension, corresponding to , has a unique positive global minimum, after which it is monotonically increasing with the same high-frequency behavior: see Figure 1. Concerning its relation to the full water wave equations, see , where there the author studies the accuracy of (3) in modeling real-world experiments of waves on shallow water.
In the full gravity-capillary wave problem with large surface tension (i.e. ) there exist subcritical
1.3. Formal computations and the main results
A routine nondimensionalization of (3) converts it to
where is the Fourier multiplier operator with symbol
We will henceforth be working with this version of the system. Substituting the traveling wave ansatz into (3) yields, after one integration the nonlocal profile equation
We are interested in long wavelength/small amplitude solutions of (6). Consequently we expect the wave speed to be close to the long wave speed , which in the nondimensionalized problem is exactly one. And so for , we make a “long wave/small amplitude/nearly critical” scaling of (6) by setting
In the above we have made the (convenient) choice
where is a Fourier multiplier with symbol .
For we have the expansion
With the usual Fourier correspondence of and , the above indicates the following formal expansion:
Therefore the (rescaled) profile equation (8) formally looks like
Putting , it follows that the solution satisfies the ODE
We immediately recognize (12) as the profile equation associated with solitary wave solutions of the (suitably rescaled) KdV equation (1). In particular, (12) admits a unique non-trivial even solution in given by
Note that is positive when and negative when , corresponding to solitary waves of elevation and depression, respectively.
Our main goal is to analyze to how deforms for . The main difficulty in the analysis is that the expansion (9) is not uniform in and, as a consequence, that the ODE (12) is necessarily singularly perturbed by the terms in (11).
It turns out that when there is a straightforward way to “desingularize” the problem. The main observation is that the multiplier for the operator is non-zero for all wave numbers when (i.e. is subcritical) when . Therefore the linear part of (6) can be inverted. Doing so and then implementing the scaling (7) results in a system which is not singularly perturbed in and one can use the implicit function theorem to continue the solution to . In the recent paper by Stefanov & Wright  this strategy (which was inspired by [19, 20]) was deployed for a class of pseudodifferential equations which includes (6) when . Their main result can be directly applied here. We explain this in greater detail in Section 3 but, for now, here is our result:
There exist subcritical solitary waves of depression for (4) when the capillary effects are strong. Specifically, for all and all sufficiently close to zero, there exists a small amplitude, localized, smooth, even function such that
solve (6). For any there exists such that
is the unique function with the aforementioned properties.
On the other hand, when a similar desingularization will not work. In Figure 1, note that when and (i.e. is supercritical) there is a unique at which
Thus cannot be inverted; the situation becomes more complicated. What occurs is that when the main pulse , through a sort of weak resonance, excites a very small amplitude periodic wave with frequency close to . The end result is a generalized solitary wave as described above. See Figure 2 for a sketch of the solution. Our proof is modeled on the one devised by Beale in  to study traveling waves in the full gravity-capillary problem (and which has subsequently been deployed to study generalized solitary waves in other contexts in    ). The proof is found in Section 4. Here is our result:
There exist supercritical generalized solitary waves for (4) when the capillary effects are weak. Specifically, for all and all sufficiently close to zero, there exist smooth, even functions and such that
solve (6). The functions and have the following properties.
is an exponentially localized function of small amplitude. In particular, there is a constant such that for all there exists for which
is a periodic solution of (6) whose frequency is approximately and whose amplitude is small beyond all algebraic orders. Specifically, there is a constant such that the frequency of lies in the interval and for all there is a constant for which
The waves constructed in Theorem 2 consist of a localized cure connecting two asymptotically small oscillatory end states. In addition to the existence result presented above, we also discuss how this rough “decomposition” can be combined with recent work  to provide insight into the possible stability of these waves. While we do not arrive at any definitive stability results here, we hope our study spurs additional work.
We also point out the work , where the author uses direct variational arguments to prove the existence of localized solutions to a large class of pseudo-partial differential equations that contains (4) for all values of . In the case of small surface tension, however, the fact that his waves have phase speeds slightly less than the global minimia of the phase speed (hence, are necessiarly subcritical with respect to the long-wave phase speed ). Labeling the (strictly positive) frequency where the global minima of is achieved, it follows that the localized waves constructed in  are in fact modulated solitary waves, taking approximately the form for some localized function .
The authors would like to thank Mats Ehrnström, Mark Groves, Miles Wheeler, Atanas Stefanov and Mathias Arnesen for useful conversations about this work. The work of M.A.J. was partially supported by the NSF under grant DMS-1614785. The work of J.D.W. was partially supported by the NSF under grant DMS-1511488.
Here we specify the notation for the function spaces we will be using along with some other conventions.
2.1. Periodic functions
We let be the usual “” Sobolev space of -periodic functions. We denote and . Put
By we mean the space of times differentiable -periodic functions and is the space of smooth -periodic functions.
2.2. Functions on :
We let be the usual “” Sobolev space of functions defined on . For put
These are Banach spaces with the naturally defined norm. If we say a function is “exponentially localized” we mean that it is in one of these spaces with .
Put , and and denote
2.3. Spaces of operators
For Banach spaces and we let be the space of bounded linear operators from to equipped with the usual induced topology.
2.4. Big notation:
Suppose that and are positive quantities (like norms) which depend upon the smallness parameter , the regularity index , the decay rate and some collection of elements which live in a Banach space .
When we write “” we mean “there exists , , , such that for all , , and .” In particular, the constant does not depend on anything.
When we write “” we mean “there exists , , such that for any there exist such that when , and .” In this case, the constant depends on but nothing else.
When we write “” we mean “there exists , , such that for any there exists such that when , and .” The constant depends on but nothing else.
Lastly, when we write “” we mean “there exists , , such that for any and there exist such that when and .” That is, the constant depends on and .
2.5. Fourier analysis:
We use following normalizations and notations for the Fourier transform and its inverse:
3. Solitary waves of depression when .
The following theorem on the existence of solitary waves in a certain class of pseudodifferential equations was proved in :
Suppose that there exists such that is (that is, its second derivative exists and is uniformly Lipschitz continuous) and satisfies
Moreover suppose that is even and there exists which has the following properties:
is (that is, its third derivative exists and is uniformly Lipschitz continuous) for .
Let be the Fourier multiplier operator with symbol . Then there exists , so that for every , there is a solution of
of the form
The function satisfies the estimate
This theorem can be applied directly to (6) when . Specifically, let , , , and . Then (6) is transformed into (18). Clearly meets the required hypotheses in Theorem 3. Given the graph of in Figure 1, it is easily believed—and even true—that meets all conditions (a)-(c) when . Thus we get the conclusions of the theorem. Unwinding the very simple rescalings gives us most of Corollary 1. Note that Theorem 3 only tells us that the solutions are in whereas Corollary 1 tells us they are smooth. In this problem, however, we have the additional information that grows like for large and hence is “like” . With this, a straightforward bootstrapping argument demonstrates that the solutions are smooth. We omit these details.
4. Generalized solitary waves when .
In this section we prove Theorem 2. Throughout we fix and we will, for the most part, not track how quantities depend on this quantity.
4.1. A necessary solvability condition
We begin our proof of Theorem 2 by doing something that is doomed to fail. Nevertheless, we believe that understanding the mechanism behind this failure is an important step in the journey to the proof of Theorem 2. Throughout, (a regularity index) is fixed but arbitrary and (a decay rate) is taken to be sufficiently small.
To this end, we first attempt to construct solutions of the nonlocal profile equation (8) for of the form
where is to be some small, smooth, even
Note that is simply the linearization of (8) about the trivial solution . is obviously quadratic in the unknown , thus we obtain the following estimates via Sobolev embedding when :
Above, by we simply mean the quantity evaluated at a function instead of at .
As for , it is a small forcing term. From its definition and that fact that solves (12) we see that
Then the formal expansion of in (10) indicates that . This argument can be made rigorous by way of Fourier analysis:
There exists so that for any and we have
The proof is in the Appendix.
To attempt to solve the nonlinear problem (20), we first consider the solvability of the nonhomogeneous linear equation
Recall from (13) that for and there exists unique such that . This implies that the symbol
associated to the linear operator satisfies
From the above considerations it is easy to conclude that
Taking the Fourier transform of (24) and evaluating at implies that (24) is only solvable provided that and the forcing satisfy . For a generic , it follows that the single unknown is required to solve two equations, hence the linear problem (24) is overdetermined. Consequently, the above method of constructing a localized solution of (8) of the form (19) fails.
4.2. Beale’s method
Beale encountered nearly the same obstacle encountered in Section 4.1 in his work on the full gravity-capillary water wave problem . In his investigation, he made the remarkable observation that just as the special frequency causes difficulties at the linear level, it also points to a way out. Indeed, observe that the lack of solvability of the linear forced equation (24) stems from (26) which, when written on the spatial side, simply states that the linear problem has a solution of the form Beale used this observation to motivate a refinement of the ansatz (19) that incorporates a family of small amplitude, nonlinear periodic traveling waves associated to the governing profile equation which are roughly given by where . By using the amplitude of this oscillation as an additional free variable, he was able to overcome the above difficulties.
In order to adapt Beale’s method to the present case, we begin by recalling that, for each fixed , the nonlinear profile equation (8) admits a family of small amplitude, spatially periodic solutions with frequencies close to . Indeed, the following result follows from the analysis of :
Fix . There exists , and a mapping
with the following properties:
For each , the function solves (8) for all .
There exists such that and imply
For all there exists such that and imply
Following Beale, we refine the ansatz (19) by including one of the above small amplitude waves. Specifically, introducing the notation
we attempt to construct solutions of the profile equation (8) for of the form
where here , and are as before. Note that the term is clearly nonlinear in the unknowns and . The term however has an term coming from the fact that . Incorporating this additional linear term on the left hand side of leads to the nonlinear equation
where here we have
There exists and such that, for all , and we have
As for , it is roughly . Specifically:
There exists and such that, for all , and we have
In both (34) and (35) above, simply represents the quantity evaluated at and . An important feature of these estimates is that there is a mismatch in the decay rates of the pieces in the estimate in (35)(ii): specifically, on the left we measure in but the right requires and to be in with . In particular, the constant diverges as . We provide the justification for Lemmas 6 and 7 in the Appendix.
Our goal is now to resolve the nonlinear equations (33) for and . As in Section 4.1, we will proceed by first considering the solvability of the associated nonhomogeneous linear equation. After we have shown that this can be continuously solved, we solve the full nonlinear equation (33) through iteration.
4.3. The linear problem
The left hand side of (33) is linear in and . We claim it is a bijection in an appropriate sense. Specifically we have the following linear solvability result.
There exists and for which the following hold when , and . There are linear maps
if and only if
Moreover these maps are continuous and satisfy the estimates
It is important to note that the size of is directly related to the (Sobolev) smoothness of the forcing function . This observation will be important in our coming work.
to hold. Since by Theorem 5, we can calculate
Note that since is positive, we know . Moreover, the analyticity of and the fact that implies is exponentially small in . Consequently and are bounded uniformly in for . It follows that we can solve (37) for explicitly in terms of and as
By construction, for all
To prove this, one needs the Riemann-Lebesgue estimate
There exists and such the following holds for all , and . Suppose that and . Then there exists a unique , which we denote by , such that . Finally, we have
At first glance, it is not entirely clear that we have made progress towards our goal of solving for , as we have to figure out how to invert the operator . To make this step requires a critical feature of : it is small perturbation of . Specifically, if we put
then we can establish the following result.
There exists and such that for all , and we have
Lemma 10 provides a meaningful and rigorous version of (45) and as such represents one of the keys of our analysis. Indeed, as in our discussion directly above the statement of Theorem 2, the is a singular perturbation of the operator , and resolving this singular limit is one of the main technical difficulties faced in the present study.