Page 1

arXiv:1111.3637v1 [math.AP] 15 Nov 2011

Particle paths in small amplitude solitary waves with negative vorticity

Ling-Jun Wang

College of Science, Wuhan University of Science and Technology, Wuhan 430081, China

Abstract We investigate the particle trajectories in solitary waves with vorticity, where the vorticity is

assumed to be negative and decrease with depth. We show that the individual particle moves in a similar

way as that in the irrotational case if the underlying laminar flow is favorable, that is, the flow is moving in

the same direction as the wave propagation throughout the fluid, and show that if the underlying current

is not favorable, some particles in a sufficiently small solitary wave move to the opposite direction of wave

propagation along a path with a single loop or hump .

Keywords:

2010 MSC: 35J65, 34C05, 76B15

particle paths, solitary water waves, negative vorticity

1. Introduction

The water-wave problem concerns the gravity-driven flow of a perfect fluid of unit density; the flow

is bounded below by a rigid horizontal bottom {Y = −d} and above by a free surface {Y = η(X,t)}, where

η depends upon the horizontal spatial coordinate X and time t. Steady waves are waves which propagate

from left to right with constant speed c and without change of shape, so that η(X,t) = η(X − ct). Solitary

waves are steady waves which have the property that η(X − ct) → 0 as X − ct → ±∞. We consider in this

paper the particle trajectories in a fluid as a solitary wave propagates on the free surface, assuming that the

flow admits a negative vorticity decreasing with depth.

There have been a series of works concentrating on the study of solitary waves, in the setting of both

irrotational flows [1, 2, 3, 6] etc., and rotational flows which become active only in the last few years. One

of the interests in the above works is the description of individual particle path in the fluid. In irrotational

flow, particle paths underneath a solitary wave are investigated in [4, 6] , both for the smooth solitary wave

and the solitary wave of greatest height for which the crest is a stagnation point. It was shown in [6] that

in an irrotational solitary water wave, each particle is transported in the wave direction but slower than the

wave speed; as the solitary wave propagates, all particles located ahead of the wave crest are lifted while

those behind have a downward motion, with the particle trajectory having asymptotically the same height

above the flat bed. For rotational flow, some results on particle paths under periodic waves are obtained

(cf. [10, 9, 15] for instance), following a series of works on the corresponding results in irrotational case (see

[5, 8, 12] etc).

Recently, rigorous existence results on small-amplitude solitary waves with arbitrary vorticity distri-

bution were obtained in [13] and in [11], using generalized implicit function theorem of Nash-Moser type

and spatial dynamics method, respectively. The solitary waves established in [13, 11] are of elevation and

decays exponentially to a horizontal laminar flow far up- and downstream. The study of solitary waves

of large-amplitude with an arbitrary distribution of vorticity remains open. So this arises the question of

particle paths in a rotational small-amplitude solitary wave. Following the pattern in [6] for irrotational case,

we prove in this work the corresponding results on particle paths in rotational solitary waves by using the

Email address: wanglingjun@wust.edu.cn (Ling-Jun Wang)

Page 2

properties of solutions obtained in [13]. We consider in this work only solitary waves with negative vorticity,

however with modifications our arguments can be applied also for positive vorticity.

Precisely we show that if the vorticity is negative and increasing from bottom to surface of the flow,

and if the underlying current is favorable, i.e., is moving throughout the fluid in the same direction as the

wave, then as time goes on the particle moves similarly as in the irrotational case [6]. We also show that

if the underlying current is not favorable, then in solitary waves with sufficiently small amplitude, some

particles above the flat bed move to the opposite direction of wave propagation along a path with a single

loop or a single hump. Note that this single-loop kind of path does not exist in the irrotational case. In [13],

as in most of the works on waves with vorticity, the author considered only waves that are not near breaking

or stagnation, i.e., the speed of an individual fluid particle is far less than that of the wave itself throughout

the fluid domain. We consider only such waves as well. We do not consider the case of wave with stagnation

which is however studied in [6] in the irrotational case.

The paper is organized as follows. In Section 2 we present the mathematical formulation for the

solitary waves and recall from [13] some useful properties of the established solitary wave solutions. Section

3 contains some conclusions on the vertical and horizontal velocity that are relevant for our purposes. The

main result is presented and proved in Section 4. In the final section we give two examples which lie in our

settings.

2. Preliminaries

We first describe the governing equations for rotational solitary water waves and then recall some

properties available on their solitary wave solutions.

2.1. The governing equations for rotational solitary water waves.

Choose Cartesian coordinates (X,Y ) so that the horizontal X-axis is in the direction of wave prop-

agation and the Y -axis points upwards. Consider steady waves traveling at constant speed c > 0. In the

frame of reference moving with the wave, which is equivalent to the change of variables (X −ct,Y ) ?→ (x,y),

we use

Ωη= {(x,y) ∈ R2: −d < y < η(x)},

to denote the stationary fluid domain and (u(x,y),v(x,y)) to denote denote the velocity field, and define

the stream function ψ(x,y) by ψ(x,η(x)) = 0 and

0 < d < ∞,

ψy= u − c,ψx= −v. (1)

Consider also only waves that are not near breaking or stagnation, so that ψy(x,y) ≤ −δ < 0 in¯Ωηfor some

δ > 0, which implies that the vorticity ω = vx− uyis globally a function of the stream function ψ, denoted

by γ(ψ); see [7]. The solitary-wave problem is then, for given p0< 0 and γ ∈ C1([0,−p0];R), to find a real

parameter λ, a domain Ωηand a function ψ ∈ C2(¯Ωη) such that

ψy< 0,(x,y) ∈¯Ωη,

(x,y) ∈ Ωη,

y = η(x),

(2)

△ψ = −γ(ψ),

|∇ψ|2+ 2gy = λ,

(3)

(4)

ψ = 0,y = η(x),(5)

ψ = −p0,

η(x) → 0

y = −d,

as|x| → ∞,

(6)

(7)

ψx(x,y) → 0as |x| → ∞ uniformly for y.(8)

2

Page 3

Here g > 0 is the gravitational constant of acceleration,

p0=

?η(x)

−d

ψy(x,y) dy < 0

is the relative mass flux (independent of x), and the boundary conditions (7) and (8) express that the wave

profile approaches a constant level of depth and the flow is almost horizontal in the far field, respectively.

Moreover we require that the nontrivial solitary wave is of positive elevation, i.e., η(x) > 0 for all x ∈ R. It

is therefore symmetric about its single crest and admits a strictly monotone wave profile on either side of

this crest (see [14]). Assuming the wave crest is located at x = 0, the solitary-wave problem (2)-(8) is thus

supplemented with the symmetry and monotonicity conditions

ψ(−x,y) = ψ(x,y),η(−x) = η(x), and η′(x) < 0 for x > 0. (9)

We refer to [13, 7] for more details on the derivation of the system (2)-(8).

2.2. Rotational solitary water waves.

We collect some properties of the solitary wave solutions established in [13]. Let

Γ(p) =

?p

0γ(−s)ds and Γmin= min

p∈[p0,0]Γ(p) ≤ 0. (10)

Given p0< 0 and γ ∈ C1([0,−p0];R), for each λ ∈ (−2Γmin,∞) the system (2)-(8) admits a trivial solution

pair (η(x),Ψ(y)) defined on¯Ω0, where η(x) ≡ 0, the stream function Ψ(y) is x-independent and is the inverse

of the function

y(Ψ) =

?−Ψ

p0

dp

?λ + 2Γ(p)− d,

and the fluid domain

Ω0= {(x,y) ∈ R2: −d < y < 0},d =

?0

p0

dp

?λ + 2Γ(p).

The corresponding relative horizontal velocity and vertical velocity are thus given by

U(y) − c = Ψy(y) = −

?

λ + 2Γ(−Ψ(y)),V (x,y) = −Ψx(y) ≡ 0. (11)

Note that Ψ(0) = 0 and Ψ(−d) = −p0. Thus

U(0) = c −

√

λ andU(−d) = c −

?

λ + 2Γ(p0).

Throughout this paper, we adopt the terminology in [8] to say this underlying trivial flow is favorable if

U(y) ≥ 0 for all y ∈ [−d,0], is adverse if U(y) < 0 for all y ∈ [−d,0], and is mixed if U(y) changes sign. Note

that favorable flow moves in the same direction as the wave propagation (i.e., to the right), while adverse

flow moves in the opposite direction of the wave propagation.

To ensure the existence of nontrivial small amplitude solitary waves, the parameter λ must be chosen

to satisfy λ > λcbut close to λc, where λc> −2Γminis the unique solution of

?0

p0

dp

(λc+ 2Γ(p))3/2=1

g.

(12)

For each such a given λ and for given p0 < 0, it was shown in [13] that there exists a nontrivial small

amplitude solitary-wave solution pair (η(x),ψ(x,y)) to (2)-(8) defined on¯Ωη, with η(x) satisfying

|η(x)| + |η′(x)| + |η′′(x)| ≤ (λ − λc)r for all x ∈ R

(13)

and for some constant r > 0 independent of λ, and the corresponding horizontal velocity satisfying the

following properties:

3

Page 4

(P1) u(x,η(x)) → c −√λ = U(0),

(P2) u(x,y) → U(y)

as |x| → ∞;

uniformly for as |x| → ∞y.

The first property is obvious since by (4), (7) and (8), we have (u(x,η(x)) − c)2= ψ2

|x| → ∞, which equivalently gives (P1) as we have assumed u(x,y) − c = ψy(x,y) < 0 in¯Ωη, while the

property (P2) can be deduced from the construction of solitary wave solutions; see [13]. Indeed for λ = λc+ε

with ε > 0, there exists a function wε(q,p), whose derivatives with respect to (q,p) up to order 2 tend to 0

uniformly for p as |q| → ∞, such that the horizontal velocity is determined by

y(x,η(x)) → λ as

u(x,y) = c −

1

(λ + 2Γ(−ψ(x,y))−1/2+ εwε

p(√εx,−ψ(x,y))

,

where wε

p(q,p) denotes differentiation in the p-variable.

We conclude this section by recalling some properties of streamlines. Due to ψy< 0 throughout¯Ωη,

we have that for all p ∈ [−p0,0] the streamline

{(x,y) : ψ(x,y) = −p}

is a smooth curve y = σp(x). Note that

σ0(x) = η(x),σp0(x) = −dand σ′

p(x) = −ψx(x,σp(x))

ψy(x,σp(x)). (14)

Observing the fact that

that for each fixed y0∈ [0,η(0)] the streamline y = σp(x) with p = −ψ(0,y0), passing through the point

(0,y0), has an asymptote y = l(y0) as |x| → ∞, with l(η(0)) = 0 and l(−d) = −d.

1

|ψy|=

1

c−uis bounded and that ψx(x,y) → 0 uniformly for y as |x| → ∞, we deduce

3. Vertical and horizontal velocity

We first divide the fluid domain Ωηinto two components

Ω−= {(x,y) ∈ R2: x < 0,0 < y < η(x)} andΩ+= {(x,y) ∈ R2: x > 0,0 < y < η(x)},

and denote their boundaries by

S−= {(x,y) ∈ R2: x < 0,y = η(x)},B−= {(x,y) ∈ R2: x < 0,y = −d},

respectively

S+= {(x,y) ∈ R2: x > 0,y = η(x)},

Lemma 3.1. Suppose that γ′(p) ≤ 0 for all p ∈ [0,|p0|]. Then

B+= {(x,y) ∈ R2: x > 0,y = −d}.

(a) v(x,−d) = 0 for all x ∈ R, and v(0,y) = 0 for y ∈ [−d,η(0)] ;

(b) v(x,y) < 0 if (x,y) ∈ Ω−∪ S−, and v(x,y) > 0 if (x,y) ∈ Ω+∪ S+;

(c) vy(x,−d) < 0 if x < 0, and vy(x,−d) > 0 if x > 0;

(d) vx(0,y) > 0 for y ∈ (−d,η(0)).

4

Page 5

Proof. Since v = −ψx, the first result follows from (6) and the symmetry property ψ(−x,y) = ψ(x,y) in (9).

Next we only prove the lemma for x > 0 and the results for x < 0 can be proved similarly. Differen-

tiating (5) with respect to x gives v = −ψx= ψyη′(x), from which follows v > 0 for (x,y) ∈ S+as ψy< 0

and η′(x) < 0 for x > 0 in view of (2) and (9). To prove v > 0 in Ω+, we assume first on the contrary that

there exists a point (x0,y0) ∈ Ω+such that v(x0,y0) = −ε < 0. Then we can find a bounded domain

Ω+,k= {(x,y) ∈ R : 0 < x < k,−d < y < η(x)},k ∈ R+,

such that (x0,y0) ∈ Ω+,k. Moreover in view of (8) we can choose k sufficiently large so that v(k,y) > −ε

for y ∈ [−d,η(k)]. This means that v attains its minimum at the interior point (x0,y0) of Ω+,k, which

contradicts to the strong maximum principle applied to v on the domain¯Ω+,k, as v satisfies △v+γ′(ψ)v = 0

with γ′(p) ≤ 0 by differentiating (3). Therefore we have v ≥ 0 in Ω+. If v = 0 at a point (x0,y0) of Ω+.

Again we can choose a bounded domain Ω+,ˆkcontaining (x0,y0). Then v ≥ 0 on the boundary of Ω+,ˆk. We

can thus apply the strong maximum principle on Ω+,ˆkonce more to conclude that v ≡ 0 on¯Ω+,ˆk, which

contradicts v > 0 on the half surface S+. This proves v > 0 in Ω+.

Since v attains its minimum in¯Ω+on the half bed B+and on the crest line x = 0, Hopf’s maximum

principle ensures that vy> 0 on B+and vx> 0 on {(0,y);−d < y < η(0)}, completing the proof.

The above lemma combined with (14) and the fact that ψy< 0 gives

Corollary 3.2. The streamline y = σp(x) with p ∈ (p0,0] satisfies that σ′

for x > 0.

p(x) > 0 for x < 0 and σ′

p(x) < 0

Lemma 3.3. Suppose that γ(p),γ′(p) ≤ 0 for all p ∈ [0,|p0|], that 0 < λ − λc< ε with ε small such that

nontrivial solitary wave exists, and that c ≥

?λ + 2Γ(p0). Then u(x,y) > 0 for x ∈¯Ωη.

Proof. We assume throughout this proof that γ(p) ?≡ 0 for p ∈ [0,|p0|], since we already know that for

γ(p) ≡ 0, i.e., the irrotational case, u(x,y) > 0 in¯Ωη; see [6]. Recall that U(y) is the horizontal velocity

of the trivial laminar flow. Thus U′(y) = Ψyy(x,y) = −γ(Ψ(y)) ≥ 0, and there exists an interval I such

that the strict inequality holds for y ∈ I since γ(p) ?≡ 0 by assumption. This combined with (11) and the

assumption c ≥

?

It was proved in [16] that if γ(p) ≤ 0 for all p ∈ [0,|p0|], then

d

dxu(x,η(x)) ≥ 0

?λ + 2Γ(p0) gives

U(y) ≥ U(−d) = c −

λ + 2Γ(p0) ≥ 0 for y ∈ [−d,0], andU(0) > 0.

for x < 0, and

d

dxu(x,η(x)) ≤ 0 for x > 0.(15)

In other words, along the free surface u increases from x = −∞ to the crest x = 0, and thereafter it is

decreasing. On the bottom B = {(x,y) ∈ R2: y = −d}, we have, in view of ux= −vy= ψxyand Lemma

3.1-(c),

ux(x,−d) > 0for x < 0, andux(x,−d) < 0 for x > 0. (16)

Then the fact that U(−d) ≥ 0 and U(0) > 0 together with the monotonicity properties (15) and (16) yields

u > 0 on the free surface and on the bottom. Differentiating (3) with respect to y yields that u satisfies

△u + γ′(ψ)u = γ′(ψ)c ≤ 0.

Finally considering U(y) ≥ 0 for y ∈ [−d,0] and the properties (P1)-(P2), we can deduce u > 0 in¯Ωη by

using the strong maximum principle as in the proof of Lemma 3.1-(b), completing the proof.

5

Page 6

The above lemma considers the case c ≥

remaining cases, we can only derive some conclusions on the horizontal velocity u for some special vorticity

function γ and for λ sufficiently close to the corresponding λcdefined in (12). To be exact, we prove

?λ + 2Γ(p0), that is the underlying flow is favorable. For the

Lemma 3.4. Let λcbe determined in (12) by the vorticity function γ(p) with p ∈ [0,|p0|].

(a) If γ′(p),γ′′(p) ≤ 0 for p ∈ [0,|p0|] and γ(0)√λc > −g, then there exists ε1 > 0 such that for λ ∈

(λc,λc+ ε1), we have

ux> 0 for (x,y) ∈ Ω−, and ux< 0 for (x,y) ∈ Ω+;

(b) If γ(p) < 0 for all p ∈ [0,|p0|], then there exists ε2> 0 such that for λ ∈ (λc,λc+ ε2), we have

uy> 0 for (x,y) ∈ Ω.

Proof. (a) Similar results have been obtained for periodic waves in [9], so we will adapt the proof there to

our present solitary wave case. We prove only the result for (x,y) ∈ Ω+and the rest can be proved similarly.

Note it suffices to prove that the stream function ψ satisfies

ψxy< 0 for (x,y) ∈ Ω+.

Since ψ(x,η(x)) = 0, we have ψx= −η′ψyon the surface, which combined with (4) gives

?

Differentiating the above equality and ψx= −η′ψyalong the surface gives respectively

?

1 + η′2

ψy(x,η(x)) = −

(λ − 2gη)/(1 + η′2).

ψxy+ η′ψyy= −∂x

λ − 2gη

andψxx+ η′ψyy+ 2η′ψxy+ η′′ψy= 0, on y = η(x).

Moreover on the surface one has

ψxx+ ψyy= −γ(0).

Combination of the above three equalities gives

ψxy(x,η(x)) =η′?g(1 − η′4) + 2η′′(λ − 2gη) + γ(0)√λ − 2gη(1 + η′2)3/2?

In view of (13) and γ(0)√λc> −g, we have when λ is sufficiently close to λcthat

g(1 − η′4) + 2η′′(λ − 2gη) + γ(0)

This gives ψxy< 0 on the surface since η′< 0 for x > 0 and√λ − 2gη(1 + η′2)5/2> 0. Since v(0,y) = 0,

we have ψxy = −vy = 0 on the line x = 0. On the bottom ψxy < 0 holds due to Lemma 3.1-(c). And

ψxy= ux→ 0 as x → ∞ since u(x,y) → U(y) as x → ∞. Finally ψxysatisfies

(△ + γ′)ψxy= −γ′′ψxψy≥ 0.

√λ − 2gη(1 + η′2)5/2

.

?

λ − 2gη(1 + η′2)3/2> 0.

The conclusion follows from the maximum principle.

(b) Recall from [13] that for λ = λc+ε, there exists a function wε(q,p) such that u(x,y) is determined

by

u(x,y) = c −

1

(λ + 2Γ(−ψ(x,y)))−1/2+ εwε

p(√εx,−ψ(x,y)), (17)

6

Page 7

where wε

{wε(q,p);0 ≤ ε < 1} satisfies that for ε small

p(q,p) stands for the differentiation of the function wε(q,p) with respect to p, and the family

sup

???∂j

q∂k

pwε(q,p)??;j + k ≤ 2,(q,p) ∈ R × [p0,0]

?

≤ r,(18)

with r > 0 some constant independent of ε. Differentiating (17) with respect to y gives

uy=

−ψy

?

−γ(ψ)(λ + 2Γ(−ψ))−3/2+ εwε

?

pp(√εx,−ψ)

p(√εx,−ψ)

?

(λ + 2Γ(−ψ))−1/2+ εwε

?2

.

Since γ(p) < 0, if we denote γmax= maxp∈[0,|p0|]γ(p), then γmax< 0 and Γmax= maxp∈[p0,0]Γ(p) > 0. As a

result, observing

−γ(ψ)(λ + 2Γ(−ψ))−3/2+ εwε

pp(√εx,−ψ) ≥ −γmax(λ + 2Γmax)−3/2+ εwε

pp(√εx,−ψ)

and

lim

ε→0

?

−γmax(λ + 2Γmax)−3/2+ εwε

pp(√εx,−ψ)

?

= −γmax(λc+ 2Γmax)−3/2

due to (18), we get, in view of −γmax(λc+ 2Γmax)−3/2> 0, that

−γ(ψ)(λ + 2Γ(−ψ))−3/2+ εwε

pp(√εx,−ψ) > 0

when ε is sufficiently small. This combined with −ψy> 0 gives uy> 0 for ε sufficiently small.

Lemma 3.5. If γ(p) < 0,γ′(p),γ′′(p) ≤ 0 for p ∈ [0,|p0|] and γ(0)√λc > −g, then for λ ∈ (λc,λc+ ε0)

with ε0= min{ε1,ε2}, along every streamline y = σp(x) with p ∈ (p0,0), the horizontal velocity u is strictly

decreasing in Ω+and strictly increasing in Ω−as a function of x.

Proof. Since

d

dxu(x,σp(x)) = ux+ uyσ′

p(x) = ux+ uy

v

u − c

due to (14), the conclusion follows immediately from Lemma 3.1 and Lemma 3.4.

4. Main result

Recalling that y = l(y0) is the streamline asymptote introduced at the end of Section 2, we are now

ready to state our main result describing the particle trajectories in a solitary wave with negative vorticity.

Theorem 4.1. Assume that γ(p) < 0,γ′(p) ≤ 0 for all p ∈ [0,|p0|], and that λ > λcsuch that solitary wave

solutions exist. Then the following results hold.

(a) Any particle above the bed reaches at some instant t0the location (X0,Y0) below the wave crest (X0,η(0));

(b) For c ≥

upwards, while for t > t0the particle moves to the right and downwards, as in Figure 1-(a); the particle

on the flat bed moves in a straight line to the right at a positive speed;

(c) If assume additionally that γ(0)√λc> −g and γ′′(p) ≤ 0 for all p ∈ [0,|p0|], then there exists ε0> 0

such that for λ ∈ (λc,λc+ ε0) and

?λ + 2Γ(p0), as time t runs on (−∞,t0), the particle above the flat bed moves to the right and

7

Page 8

(i) for

√λ < c <

flat bed; if u(x,−d) ≥ 0, depending on the relation between the asymptote y = l(Y0) and the zero

point y∗of U(y), some particles move to the right along a path with a single hump as described in

(b), some move along a single-loop path to the left, as in Figure 1-(b); if u(x,−d) < 0, there are

additionally some particles moving to the left along a single-hump path; see Figure 1-(c);

√λ, depending on the signs of u(0,η(0)) and u(0,−d), there are three possibilities for the

particles above the flat bed; see Figure 3;

?λ + 2Γ(p0), there does not exist a single pattern for all the particles above the

(ii) for c ≤

(iii) for a particle on the flat bed in both the cases (i) and (ii), if u(x,−d) ≤ 0 it moves to the left in

a straight line, while if u(x,−d) > 0 it firstly has a backward-forward pattern of motion and then

moves to the left in a straight line;

(d) The particle trajectory is strictly above the asymptote Y = l(Y0) of the streamline Y = σp(X − ct) with

p = −ψ(0,Y0).

Proof. The path (past and future) (X(t),Y (t)) of a particle with location (X(0),Y (0)) at time t = 0 is given

by the solution of the non-autonomous system

?

˙X = u(X − ct,Y ),

˙Y = v(X − ct,Y ).

Working in the moving frame x = X − ct and y = Y , we transform the above system into

?

˙ y = v(x,y).

˙ x = u(x,y) − c,

(19)

This is a Hamiltonian system with Hamiltonian ψ(x,y) in view of (1), meaning that the solutions (x(t),y(t))

of (19) lie on the streamlines. All solutions of (19) are defined globally in time since the boundedness of the

right-hand side prevents blow-up in finite time.

(a) Since in the moving frame the wave crest is assumed to be located at x = 0 in our setting, the

sign of x(t) describes the position of the particle with respect to the wave crest at time t: the particle is

exactly below the crest if x(t) = 0, is ahead of the crest if x(t) > 0 while is behind the crest if x(t) < 0. Note

that ˙ x = u(x,y) − c ≤ −δ < 0 throughout¯Ωη. This uniform upper bound on ˙ x implies that x(t) → ∓∞ as

t → ±∞ and there is a unique time t0such that x(t0) = 0. That is, to each fluid particle moving within the

water there corresponds a unique time t0∈ R so that at t = t0the particle is exactly below the wave crest,

while afterwards it is located behind the wave crest, the wave crest being behind the particle for t < t0.

In the subsequent of the proof, we always assume that the particle is located below the crest at time

t = t0at the location (X0,Y0) = (X(t0),Y (t0)), or equivalently (x(t0),y(t0)) = (0,Y0) which implies in fact

X0= ct0.

(b) Assume c ≥

for t > t0and the case t < t0can be proved similarly. Since x(t) is strictly decreasing, one has X(t) − ct =

x(t) < 0 for t > t0, which combined with Lemma 3.1-(b) implies˙Y < 0 for t > t0. Furthermore we have

˙X > 0 for all time by Lemma 3.3. That means the particle moves to the right and downwards as time

runs on (t0,+∞). The statement for particles on the bed follows from Lemma 3.1-(a) and Lemma 3.3. The

particle path for this case is depicted in Figure 1-(a).

√λ < c <?λ + 2Γ(p0), we have U(0) > 0 and U(−d) < 0, thus U(y) has a unique zero

3.4-(a) and (16) show that in¯Ω+the level set {u = 0} consists of a continuous curve C+in the moving frame.

The curve is confined between y = −d and y = y∗, and can be parameterized by x = h(y) with h′(y) > 0,

h(y) → +∞ as y → y∗, and h(−d) ≥ 0 with the equality holding when u(x,−d) = 0. In¯Ω−the level set

?λ + 2Γ(p0). For a particle located above the flat bed, we prove only the statement

(c)-(i) Since

point, say at y∗. Assume first that u(0,−d) ≥ 0. The monotonicity properties of u provided by Lemma

8

Page 9

(a)

(b)

(c)

Figure 1: Particle path with a single: (a) hump to the right; (b) loop to the left; (c) hump to the left.

u(0,−d) ≥ 0

u(0,−d) < 0

Figure 2: Particle path above the flat bed in a small solitary wave with a mixed underlying current.

{u = 0} is given by the reflection C−of the curve C+across the line x = 0. Below the curve C+and C−we

have u < 0 (including the bottom), while above and between the two curves we have u > 0. Furthermore,

in view of Lemma 3.5, if a streamline and the curve C+(rep. C−) intersect, they intersect exactly once.

Recall that (x(t),y(t)) lies on the streamline y = σp(x) with p = −ψ(0,Y0) since (x(t0),y(t0)) =

(0,Y0), and recall that y = l(Y0) is the asymptote of the streamline passing through the point (0,Y0). In

virtue of Corollary 3.2, the path (x(t),y(t)) is located below y = Y0and above the asymptote y = l(Y0) for

all time t. If l(Y0) ≥ y∗, then we have from the above arguments that u > 0 for all time t, while v < 0 if

t > t0and v > 0 if t < t0. This is the same situation as described in (b), so that the particle moves to the

right along a path with a single hump, as depicted in Figure 1-(a).

If l(Y0) < y∗, the particle trajectories above the flat bed are as shown in Figure 1-(b). Indeed, in this

case, as time t increases from −∞, the path (x(t),y(t)) intersects successively the curve C+from below at

t = t+, the vertical line x = 0 from right at t = t0, and the curve C−from above at t = t−. In the time

interval t ∈ (−∞,t+) ∪ (t−,+∞) we know that u < 0 so that in the physical variables (X,Y ) the particle

(X(t),Y (t)) moves to the left. In the time interval t ∈ (t+,t−) we have u > 0 so that (X(t),Y (t)) moves to

the right. Also we have v > 0 when t < t0so that (X(t),Y (t)) moves up, while when t > t0we have v < 0

so that (X(t),Y (t)) moves down. Thus in this case the particle above the flat bed moves to the left along a

path with a single loop.

It remains to treat the case u(0,−d) < 0. Note u(0,η(0)) > 0 in virtue of U(0) > 0 and (15). Thus,

in view of Lemma 3.4-(b), there exists a unique ˜ y ∈ (−d,η(0)) such that u(0, ˜ y) = 0. Observe ˜ y < y∗since

ux< 0 in Ω+and u(x,y) → U(y) as x → ∞. The corresponding curve C+is now located between y = y∗

and y = ˜ y, intersecting the line x = 0 at y = ˜ y. For Y0> ˜ y, the paths (X(t),Y (t)) are similar as encountered

in the case when u(0,−d) ≥ 0. While for Y0≤ ˜ y, we have (x(t),y(t)) is below C+and C−, so that u < 0 for

all the time. As a result, the particles move to the left along a single-hump path; see Figure 1-(c).

We depict all the possible trajectories in this case in Figure 2.

(c)-(ii) This case can be treated similarly as that in the above, so we omit the details and show the

particle trajectories in Figure 3.

(c)-(iii) The conclusions for particles on the flat bed follow from Lemma 3.1-(a) and (16).

(d) Observing˙Y = v(X − ct,Y ), X(t) − ct = x(t) → −∞ as t → +∞ and v(x,y) converges to 0 as

|x| → ∞ uniformly in y, we have the existence of some α such that limt→+∞Y (t) = α. Since ψ(x,y) is the

9

Page 10

u(0,−d)) ≥ 0

u(0,−d) < 0 < u(0,η(0))

u(0,−d) < 0,u(0,η(0)) ≤ 0

Figure 3: Particle path above the flat bed in a small solitary wave with an adverse underlying current.

Hamiltonian function of the system (19), we have

ψ(x(t),y(t)) = ψ(x(t0),y(t0)) = ψ(0,Y0).

Thus y(t) = σp(x(t)) with p = −ψ(0,Y0). Consequently

lim

t→+∞Y (t) = lim

t→+∞y(t) = lim

t→+∞σp(x(t)) = l(Y0).

The proof is completed.

Remark 4.2. For the path with a single loop, if we define the size of the loop by the net horizontal distance

moved by the particle between the two instants t+and t−when its horizontal velocity changes sign, that is,

X(t−) − X(t+),

then the size decreases with depth. In fact it can be computed by

X(t−) − X(t+) =

?t−

t+

dX

dtdt =

?t−

t+u(x,σp(x))dt.

Since uy> 0 and

dσp(x)

dp

= −1

ψy> 0, we have X(t−) − X(t+) decreases as p decreases.

5. Examples

In this final section we give two examples of the vorticity function which satisfy the conditions imposed

in our main theorem.

5.1. Negative constant vorticity

In the case of negative constant vorticity γ(p) = −ω0for p ∈ [0,|p0|] with ω0> 0, we have obviously

γ′(p),γ′′(p) ≤ 0. So it remains to verify γ(0)√λc> −g, or equivalently, λc< g2/ω2

λc> −2Γminis such that

?0

p0

0. Recall from (12) that

dp

(λc+ 2Γ(p))3/2=1

g

holds, where Γ(p) = −ω0p and Γmin= 0 in this case by (10). Set

F(λ) =

?0

p0

dp

(λ − 2ω0p)3/2.

Then direct computation shows that F′(λ) < 0, F(λ) → +∞ as λ → 0+, and

lim

λ→g2/ω2

0

F(λ) =1

g−

1

?g2− 2ω3

0p0

<1

g.

Thus there exists a unique λc∈ (0,g2/ω2

0) such that F(λc) = 1/g.

10

Page 11

5.2. Negative affine linear vorticity

For the vorticity γ(p) = −ap + b with a > 0 and b < 0, we have obviously γ(p) < 0, γ′(p),γ′′(p) ≤ 0

for p ∈ [0,|p0|] and

F(λ) =

p0

?0

dp

(λ + 2Γ(p))3/2=

?0

p0

dp

(λ + ap2+ 2bp)3/2,

with λ > −2Γmin= 0. Then we may verify as above that F(λ) is strictly decreasing with limλ→0F(λ) = +∞

and limλ→g2/b2 F(λ) < 0. This gives the existence of λc∈ (0,g2/b2) such that F(λc) = 1/g and consequently

γ(0)√λc> −g.

Acknowledgments

This work was supported in part by Foundation of WUST and by NSF grants of China 10901126.

References

[1] C. Amick and J. Toland, On solitary water waves of finite amplitude, Arch. Ration. Mech. Anal., 76

(1981), 9–95.

[2] T.B. Benjamin, J.L. Bona and D.K. Bose, Solitary-wave solutions of nonlinear problems, Philos. Trans.

R. Soc. London Ser. A Math. Phys. Eng. Sci., 331 (1990), 195–244.

[3] W. Craig and P. Sternberg, Symmetry of solitary waves, Comm. PDE, 13 (1988), 603–633. MR919444

(88m:35132)

[4] A. Constantin, On the particle paths in solitary water waves, Quart. Appl. Math., 68 (2010), 81–90.

[5] A. Constantin, The trajectories of particles in Stokes waves, Invent. Math., 166 (2006), 523–535.

[6] A. Constantin and J. Escher, Particle trajectories in solitary water waves, Bull. Amer. Math. Soc., 44

(2007), 423–431.

[7] A. Constantin and W. Strauss, Exact steady periodic water waves with vorticity, Comm. Pure Appl.

Math., 57 (2004), 481–527.

[8] A. Constantin and W. Strauss, Pressure beneath a Stokes wave, Comm. Pure Appl. Math., 63 (2010),

533–557.

[9] M. Ehrnstr¨ om, On the streamlines and particle paths of gravitational water waves, Nonlinearity, 21

(2008), 1141–1154.

[10] M. Ehrnstr¨ om and G. Villari, Linear water waves with vorticity: Rotational features and particle paths,

J. Differential Equations, 244 (2008), 1888–1909.

[11] M.D. Groves and E. Wahl´ en, Small-amplitude Stokes and solitary gravity water waves with an arbitrary

distribution of vorticity, Physica D: Nonlinear Phenomena, 237 (2008), 1530–1538.

[12] D. Henry, The trajectories of particles in deep-water Stokes waves, Int. Math. Res. Not., 2006 (2006),

1–13, doi:10.1155/IMRN/ 2006/23405.

[13] V.M. Hur, Exact solitary water waves with vorticity, Arch. Rational Mech. Anal., 188 (2008), 213–244,

DOI: 10.1007/s00205-007-0064-6.

11

Page 12

[14] V.M. Hur, Symmetry of solitary water waves with vorticity, Math. Res. Lett., 15 (2008), 491–510.

[15] D. Ionescu-Kruse, Particle trajectories beneath small amplitude shallow water waves in constant vorticity

flows, Nonlinear Anal-Theor., 71 (2009), 3779–3793.

[16] E. Varvaruca, On some properties of traveling water waves with vorticity, SIAM J. Math. Anal., 39

(2008), 1686–1692.

12