On the stability of solitary water waves with a point vortex

This paper investigates the stability of traveling wave solutions to the free boundary Euler equations with a submerged point vortex. We prove that sufficiently small-amplitude waves with small enough vortex strength are conditionally orbitally stable. In the process of obtaining this result, we develop a quite general stability/instability theory for bound state solutions of a large class of infinite-dimensional Hamiltonian systems in the presence of symmetry. This is in the spirit of the seminal work of Grillakis, Shatah, and Strauss, but with hypotheses that are relaxed in a number of ways necessary for the point vortex system, and for other hydrodynamical applications more broadly. In particular, we are able to allow the Poisson map to have merely dense range, as opposed to being surjective, and to be state-dependent. As a second application of the general theory, we consider a family of nonlinear dispersive PDEs that includes the generalized KdV and Benjamin--Ono equations. The stability/instability of solitary waves for these systems has been studied extensively, notably by Bona, Souganidis, and Strauss, who used a modification of the GSS method. We provide a new, more direct proof of these results that follows as a straightforward consequence of our abstract theory. At the same time, we extend them to fractional order dispersive equations.


Introduction
The persistence of localized regions of vorticity is a remarkable feature of twodimensional incompressible inviscid fluid motion. For instance, high Reynolds number flow over an immersed body may produce a wake of shed vortices outside of which the velocity field is largely irrotational. While the small-scale structure of these regions can be quite intricate, their large-scale movement is well predicted by the so-called Helmholtz-Kirchhoff point vortex model, so long as they remain sufficiently isolated. The stability of various configurations of point vortices in a fixed domain has therefore been the subject of extensive study since the early work of Poincaré [41]. In this paper, we are interested in point vortices carried by water waves. Unlike the fixed domain case, this will involve understanding the subtle dynamical implications of wave-vortex interactions. Our main results concern the orbital stability of small-amplitude solitary waves with a single point vortex.
To state things more precisely, by "water" we mean an incompressible, homogeneous, and inviscid fluid occupying a time-dependent domain t & R 2 . For simplicity, assume that at time t ! 0, t consists of the (unbounded) region lying below the graph of a function h .t; x 1 /, and above t is vacuum. This is a free boundary problem, in the sense that is not prescribed but evolves dynamically.
Let v h v.t; ¡/ t 3 R 2 denote the fluid velocity at time t ! 0. The vorticity is defined to be the quantity with each of these holding in the sense of distributions. Here p h p.t; ¡/ t 3 R is the pressure and g > 0 is the acceleration due to gravity. We consider the finite excess energy case where v.t/ P L 1 loc . t / L 2 . t n U t / for every neighborhood U t Q x x.t/. The motion of the point vortex is taken to be governed by the Helmholtz- where the subtracted term is the velocity field generated by the point vortex. Thus (1.2b) states that the vortex center does not self-advect, but rather is transported only by the irrotational part of the fluid velocity field. Finally, the evolution of the free boundary is coupled to that of the fluid by the requirements that (1.2c) @ t h . @ x 1 ; 1/ ¡ v; p h b@ x 1 @ x 1 h@ x 1 i ; on the interface S t h @ t , and where b > 0 is the coefficient of surface tension and h¡i h p 1 g j¡j 2 is the usual Japanese bracket. The first of the requirements in (1.2c) is the kinematic condition, linking the surface to the velocity field. The second is the dynamic condition, which states that the pressure deviates from atmospheric pressure (normalized here to 0) in proportion to the signed curvature of the surface.
Point vortices have been studied in fluid mechanics for centuries. The specific model (1.2a)-(1.2b) was first proposed by Helmholtz [24] and Kirchhoff [27] for incompressible fluids in a fixed domain. Later, Marchioro and Pulvirenti (see [31] and [32, chap. 4]) offered a rigorous justification by proving that (1.2a)-(1.2b) is the limiting equation governing the motion of vortex patch solutions of the Euler equations as the diameter of the patch approaches 0. Another derivation was given by Gallay [16], who showed that the system can be obtained as the vanishing viscosity limit for smooth solutions of the Navier-Stokes equation with increasingly concentrated vorticity. The recent work of Glass, Munnier, and Sueur [18] provides a second physical interpretation: they prove that the Helmholtz-Kirchhoff system governs irrotational incompressible inviscid flow around an immersed rigid body, with a fixed circulation around the body, in the limit where the body shrinks to a point in a specific way.
The primary objective in this paper is to study the stability of steady solutions of the water wave with a point vortex problem (1.2). An existence theory for waves of this type was given by Shatah, Walsh, and Zeng [43]. The analogous problem for capillary-gravity waves in finite-depth water was recently considered by Varholm [47] and for gravity waves by Ter-Krikorov [46] and Filippov [13,14]. These are among the very few examples of exact steady water waves with localized vorticity currently available. Numerical studies of water waves with a point vortex have been carried out in [10][11][12], for example. Stated informally, our main result is as follows. First, observe that in a neighborhood of S t , the velocity field v can be decomposed as v h r g r; where .t; ¡/ is harmonic in t , and is an explicit function depending on x  A more precise version is given in Theorem 6.1. Several remarks are in order. Orbital here refers to the fact that we are controlling the distance to the family of translates of the steady wave; this is natural given the translation-invariant nature of the problem. It is also important to note that 0 above is independent of t 0 , and hence the conclusion of Theorem 1.1 is much stronger than just continuity of the solution map at . c ; ' c ; x x c /. Indeed, for a global-in-time solution, this gives orbital stability in the classical sense. The norm occurring in (1.3) represents the lowest regularity in which a local well-posedness theory has been established for irrotational capillary-gravity waves [1]. On the other hand, the norm in (1.4) is associated to the physical energy for the system, which we will discuss shortly.
Our approach is to rewrite (1.2) as an infinite-dimensional Hamiltonian system of the general form du dt h J.u/DE.u/; with u in an appropriate Banach space. Here, E is a functional (the energy), and J is a state-dependent skew-adjoint operator (the Poisson map). A similar system was established formally by Rouhi and Wright [42]; we use a slightly different version and give a rigorous proof in Section 5.2.
As the entire problem is invariant under translation, there is a conserved momentum functional P h P .u/. A natural strategy for analyzing the (orbital) stability of bound states in abstract Hamiltonian systems with symmetries is to use the energy-momentum method first introduced by Benjamin [5]. In brief, this method involves constructing a Lyapunov functional using a carefully chosen combination of E and P . Actually carrying out this argument, however, can be quite challenging.
Over three decades ago, Grillakis, Shatah, and Strauss [20] introduced a powerful machinery-now commonly referred to as the GSS method-which essentially reduced these many difficulties down to discerning the convexity or concavity of a single scalar-valued quantity called the moment of instability. Not surprisingly, this paper had an enormous impact on the field and generated a great deal of research activity. However, the hypotheses of GSS limit somewhat its applicability to infinite-dimensional Hamiltonians with more complicated structure.
For instance, they require that the operator J is surjective, and independent of the state u. But recall that the Poisson map for the Korteweg-de Vries (KdV) equation is @ x , which is not surjective in the natural class of spaces. In fact, for water waves with a point vortex (1.2), we will see that J is neither independent of state nor surjective.
There is also a somewhat practical issue with the functional analytic setting. Consider for a moment the irrotational case. GSS supposes that the Cauchy problem is globally well-posed in the energy space. But, as remarked above, the local wellposedness of the gravity water wave problem with surface tension proved by Burq, Alazard, and Zuily in [1]  where G./ is the Dirichlet-Neumann operator; see the discussion in Section 5.1. For this energy to be smooth as a functional of .; '/ in the Sobolev setting, one must have that P H 3=2g ,3 W 1;I . In effect, then, there are three levels of regularity: a rough space in which the physical energy is defined, an intermediate space where the energy functional is smooth, and a yet higher regularity space where we can hope to have well-posedness. This situation is exceedingly common in the analysis of quasilinear equations. Indeed, it is the natural by-product of so-called higher-order energy estimates, which are among the most basic and widespread tools in nonlinear PDE theory.
With that in mind, as one of the primary contributions of this paper, we introduce a new abstract stability/instability result in the spirit of GSS, but with relaxed assumptions, making it directly applicable to problems such as (1.2). Specifically, we allow for a large class of state-dependent Poisson maps J h J.u/, and essentially only require that J be injective with dense range. Moreover, the entire theory is formulated in a scale of Banach spaces, offering a simple way to accommodate gaps between the necessary regularity levels for the energy. Finally, in view of the point vortex problem, we allow the symmetry group to be merely affine.
There are a number of new assumptions and technical conditions, but the main conclusion is the same as that of GSS: stability or instability of the bound state hinges on the sign of a scalar quantity. Because of the mismatch in spaces, our results are conditional in the sense that they only hold on a time interval in which the solutions of the problem exist and their growth is controlled. Using this general theory, we are then able to address the question of stability of traveling water waves with a point vortex and prove Theorem 1.1. Finally, we also consider a further application of this same framework to KdV, and related dispersive model equations.
One of the main inspirations for this paper is Mielke's work on conditional energetic stability of irrotational solitary waves on water of finite depth with strong surface tension [34], in which he also had to modify the GSS method to deal with the mismatch between well-posedness and energy spaces. While our basic strategy is the same, we make the additional effort of formulating a general theory that also deals with instability. On a technical level, the presence of the point vortex requires a number of nontrivial modifications. Mielke's work was followed by a series of papers proving the existence and conditional stability of different families of solitary water waves by a variational approach in which the waves are constructed using the direct method of the calculus of variations as minimizers of the energy subject to the constraint of fixed momentum. The stability of the set of minimizers then follows directly from classical arguments by Cazenave and Lions. In particular, Buffoni [7] considered solitary waves on finite depth with strong surface tension. He also obtained partial results in the case of finite depth and weak surface tension, as well as in the case of infinite depth [8,9], which were later completed by Groves and Wahlén [21,22]. More recently, these authors also extended the method to solitary water waves with constant vorticity [23]. Similar to the present study, the Hamiltonian formulation is noncanonical in that case. It is likely that direct variational methods could be used also in the presence of point vortices.

Plan of the Article
In Section 2, we give a detailed description of our results regarding conditional orbital stability and instability of bound states in abstract Hamiltonian systems with symmetry. Our main result on orbital stability is Theorem 2.4, which is proved in Section 3. The unstable case is addressed in Theorem 2.6, whose proof is carried out in Section 4.
We return to the water wave with a point vortex problem in Section 5, where it is shown that (1.2) can be reformulated as an infinite-dimensional Hamiltonian system of the type covered by the general theory. In Sections 6, we characterize the spectrum of the so-called linearized augmented Hamiltonian at a solitary wave, which is used to prove our main result: small-amplitude and small vorticity symmetric solitary capillary-gravity water waves with a point vortex are conditionally orbitally stable; see Theorem 6.1.
To demonstrate the broader implications of the general theory, we consider a large family of nonlinear dispersive PDEs in Section 7. These serve as approximate models for water waves, and include both the KdV and Benjamin-Ono (BO) equations. Because the corresponding J is not surjective between the relevant spaces, these equations lie outside the GSS framework. In [6], Bona, Souganidis, and Strauss overcame this difficulty by supplementing the basic approach of GSS with a consideration of the mass. On the other hand, the general theory we develop in the present paper can be directly applied to this family of equations, meaning we are able to give a new proof of the Bona-Souganidis-Strauss theorem as a straightforward application. In fact, this also furnishes new instability results for fractional KdV; see Theorem 7.4

Formulation and Hypotheses
We will work with a scale of spaces where X is a real Hilbert space, while V and W are reflexive Banach spaces. The inner product on X will be denoted by .¡; ¡/ X , and the corresponding norm by k¡k X . Likewise, let k¡k V and k¡k W be the norms for V and W , respectively. We write X £ for the (continuous) dual of X, which is naturally isomorphic to X via the mapping I X 3 X £ taking u P X to .u; ¡/ X P X £ . We will not make this identification here, but rather use I explicitly. On the other hand, we will simply identify X ££ with X, and likewise for V and W . The pairing of X and X £ we denote by h¡; ¡i X £ ¢X , while h¡; ¡i W £ ¢W is the pairing between W £ and W ; when there is no risk of confusion, we will omit the subscript.
Intuitively, X is the energy space for the system under consideration. This is where the Hamiltonian structure will be formulated, and is the natural setting for analyzing the spectrum. On the other hand, V is a space where the conserved quantities are smooth. Finally, we think of W as a "well-posedness space", with the norm coming from higher-order energy estimates used to prove that the Cauchy problem is at least locally well-posed in time. The norm on W also plays the secondary role of allowing us to get control over V via interpolation. More precisely, we require the following: Assumption 1 (Spaces). Let X, V, and W be given as above. Assume that there exist constants P .0; 1 and C > 0 such that It is often necessary to restrict attention to some smaller subset of these spaces in order to ensure that the problem is well-defined. For example, in the case of the traveling waves with a point vortex, there must be a positive separation between the vortex center and the air-sea interface. Abstractly, we will handle these types of Suppose that y J D.J / & X £ 3 X is a closed linear operator, and that we for each u P O V have a bounded linear operator B.u/ P Lin.X/. We endow X with symplectic structure in the form of the state-dependent Poisson map which is required to satisfy a number of hypotheses.
hJ.u/v; wi h hv; J.u/wi for all v; w P D. y J /.
Remark 2.2. Note that this does not assume that J.u/ is surjective, which is a significant departure from GSS. Below, we will require something slightly stronger than that the range of J.u/ be dense in X.
The main object of interest for this work is the abstract Hamiltonian system (2.3) du dt h J.u/DE.u/; uj th0 h u 0 ; where E P C 3 .O Vs R/ is the energy functional. In addition to the energy, we suppose that there is a second conserved quantity P P C 3 .O Vs R/, which we call the momentum. In order to state what it means to be a solution of (2.3), and to work with it in a meaningful way, we need to be able to view DE.u/ and DP .u/ as elements of X £ . Assumption 3 (Derivative extension). Mappings rE; rP P C 0 .O Vs X £ / exist such that rE.u/ and rP.u/ are extensions of DE.u/ and DP .u/, respectively, for every u P O V.
We say that u P C 0 .0; t 0 /s O W / is a solution of (2.3) on the interval 0; t 0 / if (2.4) d dt hu.t/; wi h hrE.u.t//; J.u.t//wi for all w P D. y J / is satisfied in the distributional sense on .0; t 0 /, the initial condition u.0/ h u 0 is satisfied, and both E and P are conserved.
Of particular importance is the situation where the system (2.3) is invariant with respect to a symmetry group. Specifically, we assume that there exists a one-parameter family of affine maps T .s/ X 3 X, with linear part d T .s/u h T .s/u T .s/0, having the properties described below. We refer to [19] for a background on affine groups on Banach spaces. P .u/ h P .T .s/u/ for all s P R: We say that u P C 1 .Rs O W / is a bound state of the Hamiltonian system (2.3) provided that it is a solution of the form u.t/ h T .ct/U c for some c P R and U c P O W .
We will also refer to U c itself as a bound state. If T represents translation, then bound states correspond to the familiar notion of traveling waves, such as the ones we will study later. For the general setting, we take it as given that an analogous family is available: (iv) Either s U 3 T .s/U c is periodic, or lim inf jsj3I kT .s/U c U c k X > 0.
Observe that, due to (2.8) and (2.9), the energy and momentum of T .s/U c are independent of s. For a fixed parameter c, the corresponding augmented Hamiltonian is the functional E c P C 3 .V Os R/ defined by E c .u/ h E.u/ cP .u/: Assumption 5 ensures that U c P D.T H .0//, and so it follows from (2.4), (2.6), and Assumption 2 that meaning U c is a critical point of E c . Due to this observation, we can think of each bound state U c as being a critical point of the energy with the constraint of a fixed momentum, with the wave speed c arising naturally as a Lagrange multiplier. Also, differentiating (2.12) with respect to c reveals that h hDP.U c /; ¡i: Commonly in applications, the bound states sit at a saddle point of the energy.
That is, the second derivative of the augmented Hamiltonian at U c has a single simple negative (real) eigenvalue and a 0 eigenvalue generated by the symmetry group, and the rest of the spectrum lies along the positive real axis, bounded uniformly away from the origin. This is the basic setting of the problem considered in Grillakis, Shatah, and Strauss [20], and it is precisely what we will encounter in our study of water waves later. We therefore make the following hypotheses about the configuration of the spectrum for the general theory.
Assumption 6 (Spectrum). The operator D 2 E c .U c / P Lin.V ; V £ / extends uniquely to a bounded linear operator H c X 3 X £ such that: (i) I 1 H c is self-adjoint on X.
(ii) The spectrum of I 1 H c satisfies (2.14) spec .I 1 H c / h¨ 2 c © f0g c ; where 2 c < 0 is a simple eigenvalue corresponding to a unit eigenvector c , 0 is a simple eigenvalue generated by T , and c & .0; I/ is bounded away from 0.

Main Results on Stability and Instability
The central question we wish to address is whether the bound states of Assumption 6 are stable or unstable. As there is an underlying invariance with respect to the group T , it is most natural to understand stability and instability in the orbital sense. For any U P X, we call the set fT .s/U s P Rg the U -orbit generated by T . Formally speaking, U c is orbitally stable provided that any solution to the Cauchy problem that is initially close enough to the U c -orbit generated by T (in the X norm) remains near the orbit for all time. Conversely, orbital instability describes the situation where there exist solutions starting arbitrarily close to the U c -orbit that nevertheless leave some neighborhood of the orbit in finite time.
Making these concepts rigorous for the problem at hand is complicated both by the lack of a global well-posedness theory for the Cauchy problem (2.3), and especially the mismatch of the energy and well-posedness spaces. For that reason, all of our results will necessarily be conditional in that they will hold only so long as we know the solution exists, and that its growth in W is controllable.
The moment of instability, which we call d , is the scalar-valued function that results from evaluating the augmented Hamiltonian along the family of bound states: Note that because each bound state U c is a critical point of the augmented Hamiltonian, differentiating d gives the identity and differentiating once more yields where the last equality follows from (2.13).
For each > 0, let be the tubular neighborhood of radius in X for the U c -orbit generated by T . We also define for all R > 0, which collapses to a ball if the symmetry group has no affine part.
Our first result states that if d HH .c/ > 0 at a certain wave speed c P I , then U c is conditionally orbitally stable.
THEOREM 2.4 (Stability). Suppose that the above assumptions hold. If d HH .c/ > 0, then the bound state U c is conditionally orbitally stable in the following sense. For any R > 0 and > 0, there exists 0 > 0 such that, if u 0; t 0 / 3 B W R is a solution of (2.3), with initial data u 0 P U X 0 , then u.t/ P U X for all t P 0; t 0 /. Remark 2.5. As will become clear in the next section, the stability theorem holds under weaker hypotheses. Most notably, we can drop the intersection with D.T H .0/j W / in Assumption 4(viii).
In order to prove an instability result, we need to know that (2.3) can be solved at least locally around the U c -orbit. If we introduce for > 0, we mean the following: Assumption 7 (Local existence). There exists 0 > 0 and t 0 > 0 such that for all initial data u 0 P U W 0 , there exists a unique solution to (2.3) on the interval 0; t 0 /.
With the above hypothesis, we can conclude that if d HH .c/ < 0, then U c is conditionally orbitally unstable.
THEOREM 2.6 (Instability). If d HH .c/ < 0 and Assumption 7 is satisfied, then the bound state U c is orbitally unstable: There exists a 0 > 0 such that for every 0 < < 0 , there exists initial data in U W whose corresponding solution exits If X h W , we also obtain a more conventional stability result as a corollary of Lastly, let us comment on how the above results relate to the recent monumental paper of Lin and Zeng [29], which studies the dynamics of linear Hamiltonian systems under weaker assumptions on the Poisson map than ours (for instance, they allow an infinite-dimensional kernel). While this theory concerns the linear case, under some conditions it can be applied to construct invariant manifolds for nonlinear systems as well; see the work of Jin, Lin, and Zeng [25,26]. When this can be accomplished, it gives considerably more information than the conditional orbital stability/instability we obtain from Theorem 2.4 or Theorem 2.6. However, the methodology has difficulty attacking equations for which the solution map incurs a loss of derivatives, such as quasilinear problems. To overcome this, one needs the linear evolution to display sufficiently strong smoothing properties, which limits somewhat its applicability. By contrast, the framework we present here is adapted to the quasilinear setting by design and does not rely on linear estimates.

Stability in the General Setting
The purpose of this section is to prove Theorem 2.4 on the conditional orbital stability of the bound state U c under the hypothesis that d HH .c/ > 0. Our basic approach follows the ideas of Grillakis, Shatah, and Strauss, but many adaptations are required due to the more complicated functional analytic setting. Interestingly, the state dependence of J is less of an issue for this argument than one may expect.
We begin with a technical lemma which states that, in a sufficiently small tubular neighborhood U X of U c , one can find a parameter value s (depending on u) such that the distance between T .s/u and U c in the energy norm is minimized. LEMMA 3.1. If s U 3 T .s/U c is not periodic, then then exists a > 0 and a function z s P C 2 .U X s R/ such that, for all u P U X , the following holds: h kT H .0/U c k 2 X .T .s/u U c ; T HH .0/U c / X : Clearly @ s h.U c ; 0/ h 0 and @ 2 s h.U c ; 0/ h kT H .0/U c k 2 X > 0. The implicit function theorem then ensures the existence of a ball B & X centered at U c , an interval . s 0 ; s 0 /, and a C 2 map z s B 3 . s 0 ; s 0 / such that the equation @ s h.u; s/ h 0 has a unique solution s h z s.u/ P . s 0 ; s 0 / for all u P B . Thus s h z s.u/ uniquely minimizes h.u; ¡/ on . s 0 ; s 0 / for any fixed u P B .
We will only present the argument for the nonperiodic orbits as the proof for the periodic case requires only a simple modification. Assumption 5(iv) then guarantees that there exists an > 0 such that inf s!s 0 kT .s/U c U c k X ! : Let h min.=3; /. Then, if u P B and r P R are such that kT .r/u U c k X kT .z s.u//u U c k, we have kT .r/U c U c k X h kdT .r/.U c u/ g T .r/u U c k X kU c uk X g kT .z s.u//u U c k X 2ku U c k X < ; which implies that r P . s 0 ; s 0 / and hence r h z s.u/ by uniqueness. This completes the proof of parts (a) and (b) for u P B .
For part (c), note that if both u and T .r/u lie in B , then kT .z s.u/ r/T .r/u U c k X h kT .z s.u//u U c k X kT .t/u U c k X for all t P R. In particular, if we choose t h z s.T .r/u/ g r, we obtain part (c) on B by uniqueness. Moreover, as a consequence, we can proceed to extend z s to all of U X through z s.u/ h z s.T .r/u/ g r; where r is such that T .r/u P B . This is well-defined, since if both T .r/u and By Assumption 6(ii), we know that X admits the spectral decomposition X h X ¨X 0¨Xg ; where X h span f c g, X 0 h span fT H .0/U c g, and X g is the positive subspace of I 1 H c . Here we are using the fact that T H .0/U c is a generator for the kernel of I 1 H c . Observe that the restriction of I 1 H c to X g is a positive operator, in the sense that there exists an h .c/ > 0 such that (3.1) hH c v; vi ! kvk 2 X for all v P X g : The following lemma describes a version of this inequality that holds also outside X g . LEMMA 3.2. Suppose that y P X is such that hH c y; yi < 0. Then there exists a constant z > 0 such that which in particular implies that a 2 0 > 0. Now, let v be as in the statement of the lemma. Using the spectral decomposition of X, we may likewise write v h a c g p for some a P R; p P X g ; as v has no component in X 0 , by assumption. Moreover, we have 0 h hH c y; vi h a 0 a 2 c g hH c p 0 ; pi; and therefore where we have utilized (3.5).
We obtain the following as a corollary.  (3.6) and Assumption 4(v). The final bound is obtained by combining this inequality with kT .z s.v//0k X h kM.v/ U c g U c d T .z s.v//vk X g kU c k X g k W ,3X kR; where we have used that v P U X , since U X is invariant under T and that W embeds continuously into X.
We will now use Lemmas 3.2 and 3.4 to obtain the key inequality needed to prove stability. It is convenient to introduce the notation M c h fu P O V P .u/ h P .U c /g for the level set of the momentum associated with U c . LEMMA 3.5. Suppose that d HH .c/ > 0. Then, for any R > 0, there exist > 0 and > 0 such that Moreover, the assumption that d HH .c/ > 0 can be removed under the additional restriction that hH c y; M.u/ U c i h 0 for a fixed y P X such that hH c y; yi < 0.
PROOF. Let u be as in the statement of the lemma, and set v h M.u/ U c . Expanding E c in a neighborhood of U c in V, recalling that U c is a critical point, and that both the energy and momentum are conserved by the group yields We wish to apply Corollary 3.3 to obtain a lower bound for hH c w; wi. In that connection, observe that hrP.U c /; wi h 0, as w is orthogonal to N by construction. where we can eliminate w in favor of v by observing that kwk 2 X ! .kvk X jjkN k X / 2 ! kvk 2 X O kvk 3 V ¡ : In either case, the desired lower bound (3.8) follows if we can control the cubic O.kvk 3 V /-remainder in (3.9) by using the quadratic kvk 2 X . This is precisely the motivation behind Assumption 1. Indeed, (2.1) and Lemma 3.4 imply that kvk 3 V C kvk 2g X kvk 1 W C R g !. g k W ,3X kR g kU c k X / g kU c k W 1 kvk 2 X ; which enables us to absorb the remainder into the quadratic term by taking sufficiently small . Note that we can replace E c by E due to the assumption that u P M c .
We are now prepared to prove the main theorem of the section on the conditional orbital stability of the bound state U c .
PROOF OF THEOREM 2.4. Seeking a contradiction, suppose there exist R > 0, > 0, and a sequence of solutions u n 0; t n 0 / 3 B W R , with initial data u n 0 , such that kM.u n 0 / U c k X 3 0, but for which kM.u n . n // U c k X h for some n P .0; t n 0 /. Without loss of generality, we may take n to be the first time that u n exits U X . Moreover, we can shrink U X such that Lemma 3.5 applies. Together with the conservation of energy and momentum, we deduce the existence of a > 0 such that E u n

Instability in the General Setting
This section is devoted to proving Theorem 2.6 on the conditional orbital instability of U c under the hypothesis that the moment of instability satisfies d HH .c/ < 0. In contrast to Section 3, the state dependence of the Poisson map J presents a more serious technical challenge to the analysis here.

Identification of a Negative Direction
Because we do not assume that J.u/ is surjective, and because c does not necessarily lie in W , we must make further modifications to the GSS program. The next lemma shows that it is possible to find a negative direction´P W that is not only tangent to M c but also lies in the range of a restriction of J.U c /. This follows from a surprisingly simple density argument.  First we claim that there exists v P U such that .N; v/ X ¤ 0. Were this not the case, we would have U & N c , which would contradict its density in X. Without loss of generality, we may choose v such that .N; v/ X h 1. Now, let u P N c be given. By density, there exists an approximating sequence fu n g & U with u n 3 u in X. Putting w n h u n .N; u n / X v; we see that the sequence fw n g & N c U and that w n 3 u .N; u/ X v h u in X; whence U is indeed dense in N c .
By the argument above, there is a sequence f´ng & U N c such that´n 3 y in X. For n sufficiently large, Q.´n/ < 0 by continuity, and so the lemma is proved.

Lyapunov Function
In the previous subsection, we constructed a vector´in the negative cone of H c , that moreover is tangent to the fixed momentum manifold M c at U c . The strategy at this point is to use´to build a Lyapunov function for the abstract Hamiltonian system (2.3) and thereby prove instability.
In the next lemma, we follow Grillakis, Shatah, and Strauss by introducing a functional A, designed so that the corresponding Hamiltonian vector field (i) points in the direction´at U c , and (ii) is in the kernel of DP in a tubular neighborhood of U c .  With the functional A in hand, we next consider the ordinary differential equation Observe that That is, the flow of (4.8) preserves the momentum. One can interpret this lemma as follows. Because of (4.12), the flow of (4.8) leaves the momentum invariant but it may change the energy in either direction near U c . By avoiding the problematic negative direction in a suitable way, and using   where we have exploited (4.12) in evaluating P at v. Then, suppressing the dependence of on .; v/, we find @ g.; v/ h hDE c ./; @ i h hDE c ./; J./DA./i; so @ g P C 1 .N s R/ as well by Lemma 4.2(e). It therefore makes sense to compute @ 2 g.0; U c / h D 2 E c .U c /@ .0; U c /; @ .0; U c / h hH c´;´i < 0; where we have used that U c is a critical point of E c , and the last inequality is Lemma 4.1.
We also see that The final lemma we need in order to prove the instability theorem is the following:   (4.11). We also know that the flow of (4.8) conserves momentum, whence .s/ P M c for all s P . 0 ; 0 /. Finally, the proof of Lemma 4.4 shows that is C 2 and that E has a strict local maximum at 0.
The result is now obtained by a possible reparametrization.   (4.15). This shows that A u s is unbounded, but we also have jA.u/j kZk X kM.u/ U c k X kZk X k W ,3X k 0 for every u P U W 0 by the definition of A in (4.2). We have arrived at a contradiction, and must conclude that the U c -orbit is unstable.

Hamiltonian Structure for Water Waves with a Point Vortex
With our general machinery in place, we turn to the question of stability of solitary capillary-gravity waves with a submerged point vortex. The next subsection recalls how this system was formulated by Shatah, Walsh, and Zeng in [43]. In Section 5.2, we show that the problem can be rewritten once more as an abstract Hamiltonian system of the general form (2.3), and verify that the corresponding energy, momentum, Poisson map, symmetry group, and bound states meet the many requirements of Section 2.1.

Nonlocal Formulation
Consider the capillary-gravity water wave problem with a point vortex described in (1.2). In any simply connected subset of t n fx xg, the velocity v can be decomposed as where is harmonic on t and is harmonic on the subset. The latter represents the vortical contribution of the point vortex. Since the surface is a graph, we will Then is harmonic on the open set fx P R 2 x 1 ¤ x x 1 or jx 2 j < x x 2 g, and r extends to a smooth velocity field on t n fx xg. The purpose of 2 , which corresponds to a mirror vortex at x x H h .x x 1 ; x x 2 /, is to make r decay faster as jxj 3 I. Indeed, r is L 2 on the complement of any neighborhood of x For later use, we also introduce notation for the harmonic conjugate of , which takes the form h 1 2 with Note that the convention used here is that r h r c , where we recall that r c is the skew gradient introduced in (1.1).
The rationale behind splitting v according to (5.1) is that it nearly decouples the tasks of determining the rotational and irrotational parts of the velocity. Indeed, is entirely explicit given x x, which solves the differential equation (1.2b). The main analytical challenge is determining and . But for this we can proceed as in the classical Zakharov-Craig-Sulem formulation of the irrotational water wave problem: Because is harmonic, it is enough to know and the trace ' h '.x 1 / h .x 1 ; .x 1 // of on the surface. Notice that and ' then have the fixed spatial domain R. The problem can therefore be reduced to the boundary, with the rotational part rj S t being viewed as a forcing term.
On S t , we must ensure that the kinematic condition and Bernoulli condition are satisfied. Naturally, these will now involve tangential and normal derivatives of and . Here and in what follows, we will therefore make use of the shorthand expressions r c h H @ x 1 g @ x 2 ¡ S t ; r b h @ x 1 g H @ x 2 ¡ S t ; which arise when parametrizing the free surface using . Note also that we are using the convention that spatial derivatives of quantities restricted to the boundary are denoted by a prime, while @ x 1 is reserved for functions of two or more spatial variables. Exceptions will be made for certain differential operators when this does not create ambiguities.
The tangential derivative r b is simply ' H , but to express the normal derivative r c requires using the nonlocal Dirichlet-Neumann operator G./ H k .R/ 3 It is well-known that, for any k 0 > 1, k P 1=2 k 0 ; 1=2 g k 0 , and P H k 0 g1=2 .R/, the operator G./ is an isomorphism. Moreover, the mapping U 3 G./ is analytic, G./ H 1=2 .R/ 3 H 1=2 .R/ is self-adjoint, and G.0/ h j@ x 1 j, @ t x x h r.x x/ @ x 1 2 .x x/e 1 ; where, to simplify the notation, we have introduced h 1 g 2 and h . x 1 ; x 2 /. The motivation for this choice is that r x x h .
The first equation in (5.4) is simply the kinematic condition in (1.2c), while the second follows from evaluating Bernoulli's law along the free surface using the dynamic condition to replace the trace of the pressure with the (signed) curvature. Finally, the third equation is just (1.2b) in view of the splitting (5.1). Observe that @ t x x can easily be eliminated from the equation for @ t ', but we opt not to do so.

Hamiltonian Formulation
We now endeavor to rewrite (5.4) as a Hamiltonian system for the state variable u h .; '; x x/. The first step is to fix a functional analytic framework. For that, we introduce the continuous scale of spaces A local well-posedness result at this level of regularity was obtained for irrotational capillary-gravity water waves by Alazard, Burq, and Zuily [1]. While the Cauchy problem for (5.4) has not yet been studied, it is reasonable to suppose that local well-posedness will hold in the same space. In our setting, this is the minimal regularity required to have the traces of the velocity be Lipschitz on the surface.
Note that our results hold with any smoother choice of W as well. We also mention that very recently Su [45] has obtained long-time well-posedness results for gravity waves with a point vortex (corresponding to b h 0). The Gagliardo-Nirenberg interpolation inequality yields the following for our choice of spaces. Lastly, recall that for the problem to be well-defined, the surface must lie between the point vortex at x x P t and its mirror at x x H . We therefore let O h fu P X x x 2 < .x x 1 / < x x 2 g and seek solutions taking values in O W at each time.
We endow X with symplectic structure by prescribing a Poisson map. First, consider the linear operator y J D. y J / & X £ 3 X defined by with the natural domain Lin.X/ Q B.u/ h Id X g K .u/; where K .u/ P Lin.X/ is the finite-rank operator given by Thus Assumption 3 is indeed satisfied. The next lemma confirms that the Hamiltonian system for this choice of the energy and Poisson map corresponds to the water wave with a point vortex problem in (5.4). PROOF. Written out more explicitly using (5.11), the Hamiltonian system (5.20) is Using (5.18), we see that the first of Hamilton's equations is which is equivalent to the kinematic boundary condition in (5.4). Moreover, the equation for @ t ' above agrees with the corresponding one in (5.4), as the final term is simply j S t ¡ @ t x x by the equation for @ t x x.
Only the equation for the motion of the point vortex remains. Written out explicitly, we find that where is the harmonic conjugate to in t and N is the outward-pointing unit normal. Now, owing to the fact that and x 1 are harmonic on R 2 n fx x; x x H g, the final integral is equal to . x 1 x 2 x 1 x 2 /dx 1 h 0 by path independence. Here we have used that h x 1 h 0 on fx 2 h 0g.
On the other hand, we have the identity In total, then, and an essentially identical argument shows that @ t x x 1 h @ x 2 .x x/ @ x 1 2 .x x/: Recalling that and are harmonic conjugates, these two equations are equivalent to the vortex dynamics equation in (5.4).
Finally, conservation of the energy E is immediate from the fact that u is a C 1 solution of (5.20). The conservation of momentum P is simply a consequence of (2.6), which we verify below in Lemma 5.4.

Symmetry
Let T h T .s/ X 3 X be the one-parameter family of affine mappings given by (5.21) T .s/u h ..¡ s/; '.¡ s/; x x g se 1 /; s P R; representing the invariance of the underlying system with respect to horizontal translations. The linear part of the family is and ( which is certainly dense in X (cf. Lemma A.1). Finally, the conservation of energy under the group (ix) is immediate given the translation invariant nature of E in (5.12), (5.13), and (5.14).

Traveling Waves
In Theorem B.1, we prove the existence of a surface of small-amplitude traveling wave solutions of the point vortex problem, parametrized by the vortex strength and the depth of the point vortex a. For the stability analysis, however, it is important to fix , as it appears as part of the equation. We will therefore consider the families kT .s/U c U c k X ! j.se 1 ae 2 / ae 2 j h jsj; so the second option in (iv) holds.
Formally, the traveling waves on C I are stable if we can show that the moment of instability defined in (2.15) has positive second derivative. This can be shown to be the case when is small. It is important at this point to emphasize that the family C I comprises all even traveling wave solutions of (5.4) with .; '; c/ and in a neighborhood of 0, in a certain function space setting; see Appendix B for more details. Thus, the stability furnished by Theorem 6.1 applies to all even waves that are sufficiently small amplitude, slow moving, and have small enough vortex strength.
We have already addressed a number of the hypotheses of the general theory. Moreover, Lemma 5.6 shows that the family C I is formally orbitally stable for 0 < jj ( 1. The only remaining task-which is by far the most difficult-is to verify that the waves in C I lie at a saddle point of the energy with a one-dimensional negative subspace, as required by Assumption 6. Our basic approach follows along the lines of Mielke's study of the irrotational case [34], with many modifications necessitated by the presence of the point vortex.
Recall that the family of traveling waves fU c g are critical points of the augmented Hamiltonian E c h E cP . Because ' occurs quadratically in E, and hD ' E c ; 'i h R '.G./' g r c g c H /dx 1 ; we can eliminate ' by introducing The next lemma further unpacks the expression (6.3) to obtain a quadratic form representation on the energy space, in preparation for the verification of Assumption 6.
which yields the claimed operator A. and all three terms are diagonal matrices.
We can now confirm that the augmented Hamiltonian admits an extension to the energy space.
where L .v/ and A.v/ are as defined in Lemmas 6.2 and 6.3, respectively. The adjoint L .v/ £ P Lin.X 2 s X £ 1;3 / is given by for all u P V, and it is simple to verify that for all v P V 1;3 and ' P V 2 .
Using the representation for D 2 V aug c furnished by Lemma 6.5 in conjunction with the asymptotics derived in Appendix B, we are at last able to prove that Assumption 6 is satisfied.  , respectively, described in Section 1. KdV, among many other things, governs surface waves in shallow water. Benjamin-Ono models the motion of waves along the interface between two infinitely deep fluid regions in a certain long-wave regime [4,38]. In [6], Bona, Souganidis, and Strauss investigated the orbital stability and instability of solitary-wave solutions to (7.1) for P 1; 2. Their strategy relied on many of the ideas underlying the GSS method. However, as we will see below, the corresponding Poisson map J was not surjective, and hence a number of adaptations were necessary. Specifically, the authors made use of another conserved quantity-the mass u dx-requiring them to obtain estimates on the spatial decay rates of solitary waves in order to ensure the persistence of integrability. Our purpose in this section is to offer a new proof of the Bona, Souganidis, and Strauss theorem that follows directly from the stability machinery presented in Section 3 and Section 4. Because we do not appeal to the mass, no asymptotic estimates are required. Notice also that we treat "fractional" dispersive model equations for which P .1=3; 1/. Orbital stability results for P .1=2; 1/ have been obtained by Linares, Pilod, and Saut [30] and Angulo Pava [39]; we discuss the connections between these works and the present paper further below. Theorem 7.4 below gives conditional orbital instability for fKdV (p h 2) when P .1=3; 1=2/, and this appears to be new. Indeed, Linares, Pilod, and Saut observe that the Bona, Souganidis, and Strauss approach almost works in this regime, except that the tail estimates fail to hold.
While we do not pursue it here, one can also consider more general nonlinearities at the expense of some sharpness. Another interesting possible extension is to study dispersive PDEs like the Whitham equation, where in (7.1) is replaced by a Fourier multiplier with an inhomogeneous symbol.
It is also important to note that, by specializing to specific choices of and p, one can say much more. As one example, for h 2 (gKdV), Pego and Weinstein [40], Mizumachi [35], Martel and Merle [33], and Germain, Pusateri, and Rousset [17] obtain asymptotic stability results (in different topologies) for various subcritical cases p < 5. For supercritical waves p > 5, Jin, Lin, and Zeng [26] were able to completely classify the H 1 dynamics near the family of solitary waves using invariant manifold techniques. The main appeal of our approach is its relative simplicity, and the fact that it simultaneously addresses the range of dispersion strengths P .1=3; 2 and nonlinearities (7.2).

Reformulation as a Hamiltonian System
Formally, the expression inside the parentheses on the right-hand side of (7.1) is the derivative of the energy This suggests that the natural energy space is X h H =2 .R/, with the dual space X £ h H =2 .R/, and the isomorphism I X 3 X £ given by hi . The condition > 1=3 ensures the existence of admissible p, those satisfying (7.2), which in particular implies that X ,3 L pg1 .R/. Observe that E defined according to (7.3) then lies in C I .Xs R/, and that indeed DE.u/ h u u p for all u P X. We may therefore take (7.4) V h X: The local and global well-posedness of the Cauchy problem for (7.1) is still an active subject of research, and what is currently known depends considerably on and p. To state things concisely, we suppose that (7.1) is known to be locally well-posed in H s for s > s 0 h s 0 .; p/, and set At present, the best-known result when p h 2 is s 0 .; 2/ h 3=2 5=4, and hence (7.1) is globally well-posed in X when > 6=7 and p h 2; see [36,37]. This is conjectured to hold for all > 1=2, which corresponds to the L 2 subcritical case.
However, for fKdV with P .1=3; 6=7, the functional analytic setup in (7.5) will lead to a conditional stability or instability result. This is essentially what is done by Angulo Pava in [39, theorem 1.1], as well as Linares, Pilod, and Saut in [30, theorem 2.14], who treat the range P .1=2; 1/. We caution, however, that in both of these papers the definition of "conditional stability" is less conditional than ours: we require the solution to remain in the ball B W R , while they only ask for it to exist.
Next, define the Poisson map J D.J / & X £ 3 X by (7.6) J h @ x ; with domain D.J / h H 1g 2 .R/. As J is independent of state, it can be identified with y J in Assumption 2. Moreover, J is clearly injective, and skew-adjoint. The Cauchy problem for (7.1) can now be restated rigorously as the abstract Hamiltonian system (7.7) d dt hu.t/; wi h hu p u; @ x wi for all w P H 1g 2 .R/; u.0/ h u 0 ; by specializing the general system in (2.4).
The equation ( which also is of class C I .Xs R/.
The next lemma collects and expands upon these observations to confirm that the Hamiltonian formulation meets the requirements of the general theory. PROOF. Because V h X, both Assumption 1 and Assumption 3 hold trivially.
Likewise, for J defined as in (7.6), we have already verified that the relevant requirements of Assumption 2 are met. To show that the symmetry group satisfies Assumption 4 requires chasing the definitions. Both invariance and the commutativity are readily checked, as differentiation commutes with translation. Finally, the only remaining property that requires elaboration is (viii). We see that which is dense in X by the same kind of argument as in Lemma A.1.

Solitary Waves and Spectral Properties
It is well-known that the dispersive models captured by ( Q g Q h Q p : LEMMA 7.2. If Q P X is a nontrivial solution of (7.12), then the family fU c c P .0; I/g defined through (7.11) satisfies Assumption 5.
PROOF. By a standard bootstrapping argument, we have that any such solution Q lies in H r .R/ for every r ! 0. Since U c is defined by (7.11), parts (i)-(ii) are therefore immediate. Finally, lim inf jsj3I kT .s/U c U c k X h 2kU c k X > 0; so the second option of part (iv) holds.
It is also easily seen that if Q ¤ 0 solves (7.12), then Q is a critical point of the Weinstein functional J P C 2 .X n f0gs .0; I// defined by We say that a solution Q of (7.12) is a ground state if Q is not just a critical point of J but also an even, positive minimizer. Since J is invariant under scaling, the same is then true of each U c , solving (7.10), defined through (7.11). Note that the corresponding operator H c P Lin.X; X £ / is given by H c u h D 2 E c .U c /u h u pU p 1 c u g cu; which is clearly self-adjoint. We have the following result, due to Frank and Lenzmann [15], vastly generalizing earlier results for KdV [48] and BO [2,3]. LEMMA 7.3. There exists a unique ground state solution Q P X of (7.12). Moreover, for each c P .0; I/, the spectrum of the operator H c corresponding to the bound state solution U c defined by (7.11) satisfies spec I 1 H c h¨ 2 c © f0g c ; where 2 c < 0 and 0 are simple eigenvalues, and c & .0; I/ is bounded away from 0. That is, Assumption 6 is satisfied.
Of course, Q cannot be written down explicitly for most choices of and p. for BO in his original paper on the topic.

Stability and Instability
The analysis of the previous subsection confirms that the family fU c c P .0; I/g corresponding to the unique ground state Q of (7.12) that is furnished by

Appendix A Function Spaces
Define the Schwartz class S .R/ to be the set of all f P C I .R/ such that x n f .m/ .x/ P L I .R/ for all n; m P N 0 , and also the subspace S 0 .R/ of those f P S .R/ for which y f .n/ .0/ h 0 for all n P N 0 . For every s P R, we define

Appendix B Existence Theory
In this appendix, we present a slightly modified version of the existence theory for capillary-gravity waves with a point vortex due to Shatah, Walsh, and Zeng [43].
The original paper fixes the location of the vortex, which is ill-suited for us. Since appears in the Poisson map, it must be held fixed on any family of waves to which we wish to apply the general stability theory. We can obtain such families by also allowing the location of the point vortex to vary.  which is clearly an isomorphism X 3 Y , as it is a lower diagonal matrix with isomorphisms on the diagonal. Finally, the asymptotic expansions listed in (B.2) can be found by implicit differentiation.
It is possible to write the leading-order surface term 2 in terms of the so-called exponential integral E 1 . Note that in the above computations we have made repeated use of the fact that is harmonic in a neighborhood of the surface S. In particular, this implies that (C.1) r c x 1 h r b x 2 h . x 2 j S / H ; r c x 2 h r b x 1 h . x 1 j S / H : Similar identities hold for as well.