V. A. Dougalis and D. E. Mitsotakis, Solitary waves of the Bona-Smith system, in Advances in scattering theory and biomedical engineering, ed. by D. Fotiadis and C. Massalas, World Scientific, New Jersey, 2004, pp. 286–294.

SOLITARY WAVES OF THE BONA - SMITH SYSTEM

V.A. DOUGALIS AND D.E. MITSOTAKIS Department of Mathematics, University of Athens, 15784 Zographou, Greece, and Institute of Applied and Computational Mathematics, FO.R.T.H., P.O. Box 1527, 71110 Heraklion, Greece

We consider the Bona - Smith family of Boussinesq systems and show, following Toland’s theory, that it possesses solitary wave solutions for each k > 1, where k is the speed of propagation of the wave. In addition, the solitary waves are shown to be unique if k is small enough. We also make a brief computational study of the stability of these solitary waves.

1. Introduction The Bona - Smith systems are Boussinesq type systems [2], of the form ηt + ux + (ηu)x + auxxx − bηxxt = 0 ut + ηx + uux + cηxxx − duxxt = 0,

(1)

where a = 0, b = d = (3θ2 − 1)/6, c = 2/3 − θ2 , and θ > 0 a parameter such that θ2 ∈ (2/3, 1]. They approximate the Euler equations of water wave theory and model one-dimensional, two-way propagation of long waves of small amplitude when the Stokes number is O(1). In their dimensionless, unscaled form represented by the system (1), x ∈ IR is a spatial variable along the channel of propagation, t > 0 is the time, η = η(x, t) is the wave height above an undisturbed level of zero elevation, and u = u(x, t) is the horizontal velocity at height θ above the bottom. (The (horizontal) bottom corresponds to an elevation equal to −1 in these variables.) The initial-value problem for the system (1) posed for x ∈ IR, t > 0, with given smooth initial data η(x, 0) = η0 (x), u(x, 0) = u0 (x), x ∈ IR, has been studied in [3], in the case θ2 = 1. A straightforward extension of the theory of [3], yields that if η0 ∈ H 1 (IR) ∩ Cb3 (IR), u0 ∈ L2 (IR) ∩ Cb2 (IR) are 1

2

such that η0 (x) > −1 for x ∈ IR and Z +∞ p E(0) := [η02 + (θ2 − 2/3)(η00 )2 + (1 + η0 )u20 ]dx < 2 θ2 − 2/3, −∞

then, the initial-value problem for (1) has a unique classical solution (η, u) ∈ C(0, T ; H 1 (IR) ∩ Cb3 (IR)) × C(0, T ; L2 (IR) ∩ Cb2 (IR)) for each T > 0. The crucial step in the proof is to establish an a priori H 1 × L2 estimate that follows from the hypotheses initial data and the invariance of the R +∞ 2 on the 2 Hamiltonian E(t) = −∞ [η + (θ − 2/3)ηx2 + (1 + η)u2 ]dx for t > 0. We shall be particularly interested in the solitary wave solutions of the system (1). These correspond to travelling wave solutions of the form ηs (x, t) = ηs (x+x0 −kt), us (x, t) = us (x+x0 −kt), x, x0 ∈ IR, t > 0, (2) which may travel to the left or right, if the (constant) speed k is positive or negative, respectively. In (2) the univariate functions ηs (ξ), us (ξ), ξ ∈ IR, will be supposed to be positive, to have a single maximum located at ξ = 0, to be even about ξ = 0, and to decay to zero, along with their first derivatives, as ξ → ±∞. Substituting (2) into (1) and dropping the subscript s, we obtain that (u(ξ), η(ξ)) satisfy the system of nonlinear ordinary differential equations (the derivative 0 is with respect to ξ) S1 u00 + S2 u + ∇g(u, η) = 0, ξ ∈ IR, where u = (u, η)T , and ¶ µ 6 ¶ µ 6a 3 2 −k 6 − k −6b , S = S1 = 2 6 , and g(u, η) = − u η. −6d − 6c 6 − k k k

(3)

(4)

(Note that if (u, η) is a solution of (3) corresponding to some k > 0, the (−u, η) is also a solution that propagates with speed −k, i.e. to the left. Henceforth we shall take k > 0). 2. Existence of Solitary Wave Solutions The existence of solitary wave solutions of (1) was established, in the case θ2 = 1 for any k > 1, by Toland, [10]. Subsequently, in [11], by an insightful geometric proof, Toland showed that more general o.d.e. systems of the form (3) possess such symmetric ‘homoclinic’ orbits in the first quadrant of the u, η-plane. Toland’s theorem may be formulated as follows (using the notation of [4]):

3

Theorem 2.1. Let S1 and S2 be symmetric, and let g ∈ C 2 (IR2 , IR) be such that g, ∇g and the second partial derivatives of g are all zero at (0, 0). Moreover, if Q(u) = uT S1 u and f (u, η) = uT S2 u + 2g, assume that (I) det(S1 ) < 0, and there exist two linearly independent vectors u1 = (u1 , η1 )T and u2 = (u2 , η2 )T such that Q(u1 ) = Q(u2 ) = 0. (II) There exists a closed plane curve F which passes through (0, 0) such that: (i) f = 0 on F, and F \{(0, 0)} lies in the set {(u, η) : Q(u, η) < 0}, (ii) f (u, η) > 0 in the (non-empty) interior of F, (iii) F \ {(0, 0)} is strictly convex, i.e. D = fuu fη2 − 2fuη fu fη + fηη fu2 < 0 on F \ {(0, 0)}, (iv) ∇f (u, η) = 0 on F if and only if (u, η) = (0, 0). Then, there exists an orbit γ of (3) which is homoclinic to the origin. Moreover (a) (u(0), η(0)) ∈ Γ, where Γ is the segment of F not including the origin between P1 and P2 , with Pi satisfying ∇f (Pi ) · ui = 0 for i = 1, 2, (b) u, η are even functions on IR, (c) (u(ξ), η(ξ)) is in the interior of F for all ξ ∈ IR \ {0}, (d) γ is monotone in the sense that u(ξ) 6 u(s), η(ξ) 6 η(s) if ξ > s > 0. ¥ Toland’s general theory was applied by Chen, [4], to several specific examples of Boussinesq systems. It may also be applied to establish existence of solitary wave solutions of the whole family of Bona - Smith systems, i.e. for each θ2 ∈ (2/3, 1], for any k > 1. Specifically, in the case of the general Bona - Smith system (1) we have µ ¶ µ 6 ¶ µ 6 ¶ 0 −6b −k 6 − k uη . S1 = , S = , ∇g(u, η) = 2 6c 6 −6b − k 6 −k − k3 u2 Hence, S1 , S2 are symmetric and det(S1 ) = −36b2 < 0. Also Q(u, η) = − k6 (2bkuη + cη 2 ) and 6 f (u, η) = − (u2 (1 + η) + η 2 − 2kηu). (5) k It is easy to check that Q vanishes on the directions of the vectors u1 = T (1, 0)T and u2 = (1, − 2bk c ) . Hence, hypothesis (I) of Theorem 2.1 holds.

4

To verify hypothesis (II) we consider F ∗ = {(u, η) : f (u, η) = 0}. If we factor f when ηp6= −1 then f (u, η) = − k6 (1 + η)(u − u− )(u − u+ ), where u± = (kη ± η 2 (k 2 − 1 − η))/(1 + η). Therefore, if k > 1, the set {f = 0} represents a closed curve F which passes through the origin and lies on the first quadrant when 0 6 η 6 k 2 − 1 (actually F is the part of the curve F ∗ in the first quadrant). Moreover f > 0 in the interior of F since u is between u− and u+ . Furthermore F is strictly convex since D = −2(6/k)3 ((u2 − η 2 )2 + u4 η)η −1 < 0. Also, F lies in the interior of the cone K1 which is formed by the √ straight lines η = s1 u and η = s2 u, where √ s1 = k − k 2 − 1 and s2 = k + k 2 − 1. These lines are tangent to F at (0, 0). Also, the cone K1 is in the interior of the cone K, which is formed by the vectors u1 and u2 , and so F \{(0, 0)} is in the set {(u, η) : Q(u, η) < 0}, i.e. for each θ2 ∈ (2/3, 1] and k > 1 K1 ⊂ K. (It is interesting to observe that the direction of u2 tends to the line η = s2 u as θ2 ↑ 1 and k → ∞). Finally it is easy to check that, ∇f = 0 on F if and only if (u, η) = (0, 0). Note that P1 = (k − 1/k, k 2 − 1) and P2 = (2(k − 1), 2(k − 1)).

η














































P1

2

k −1

Γ (u(0), η (0)) f=0

2(k −1)

P2

2

k −1 k

Figure 1.

2(k −1)

u

Locus of possible (u(0), η(0))

In conclusion, all hypotheses of Theorem 2.1 are valid and so the general Bona - Smith system (1) possesses (single hump) solitary wave solutions. In particular, it follows from Toland’s theory that the existence of a solitary wave implies that the pair of peaks (u(0), η(0)) lies in the open segment Γ = P1 P2 of F not including the origin, cf. Fig. 1. Furthermore, since f (u(0), η(0)) = 0, it follows that the speed k of a solitary wave correspond-

5

ing to the peak (u(0), η(0)) satisfies the equation k=

µ2 (1 + η(0)) + 1 , 2µ

(6)

where µ = u(0)/η(0). The curve F and, consequently, the relation (6) are independent of the constants a, b, c, d of the particular Boussinesq system, and, in particular, independent of the parameter θ2 of the Bona - Smith systems. However, the shape of a solitary wave of an individual Bona Smith system (i.e. the dashed line in Fig. 1 representing the solitary wave (u(ξ), η(ξ)), 0 6 ξ), depends of course on θ2 . For each value θ2 in the interval (7/9, 1) one may find, following [5], one explicit solitary wave solution (us , ηs ) of the form ηs (x, t) = η0 sech2 (λ(x + x0 − kt)) , us (x, t) = Bηs (x, t),

(7)

where x0 ∈ IR is arbitrary, and the parameters η0 , k, B, λ are given in terms of θ2 by the formulas η0 = q λ=

1 2

9 2

·

θ 2 −7/9 1−θ 2 ,

3(θ 2 −7/9)

(θ2 −1/3)(θ 2 −2/3)

k=√ ,

B

4(θ 2 −2/3)

2(1−θ 2 )(θ 2 −1/3) q 2) = 2(1−θ θ 2 −1/3 .

,

(8)

3. Uniqueness of the Solitary Wave Solutions The uniqueness of the solitary wave solutions of (1) may be studied again by the methods of Toland, [12]; he established their uniqueness in the case of the θ2 = 1 system, unconditionally if u(0) 6 1, and in general if 1 < k 6 3/2 or k À 1. In the case of the general Bona - Smith system, following Toland’s √ 2+ 0.2 proof it is possible to prove uniqueness provided e.g. that < θ2 6 1 3 and 1 < k 6 kmax (θ). We outline the proof below; full details may be found in [9]. The solitary waves are solutions of the boundary-value problem  u(x), η(x), x ∈ IR     k(η − bη 00 ) = u + uη in IR   1 k(u − bu00 ) = cη 00 + η + 2 u2 in IR (9)   u0 < 0, η 0 < 0 in (0, ∞)    lim u(x) = lim η(x) = 0, u, η positive and even on IR.  |x|→∞

|x|→∞

The proof of (conditional) uniqueness of solutions of (9) will be presented as a sequence of Lemmata below. The first result, concerns some

6

needed a priori estimates and may be obtained by the fact that the peak (u(0), η(0)) must lie on the closed curve F, so that f (u(0), η(0)) = 0, and by using a classical maximum principle for two-point boundary value problems. Lemma 3.1. If (u, η) satisfies (9), then 0 < u(x) 6 u(0) 6 2(k − 1), x ∈ IR,

(10)

0 < η(x) 6 η(0) 6 k 2 − 1, x ∈ IR,

(11)

c ku(x) > − η(x) > 0, x ∈ IR, b

(12)

c ku0 (x) < − η 0 (x) < 0, x ∈ (0, ∞). b

(13) ¥

With the help of a nonlinear maximum principle, [7], we can prove as in [12], that ku(x) is bounded from above by a multiple of η(x) and vice versa for the derivatives. Specifically there holds: 2

√

c Lemma 3.2. If (u, η) satisfies (9) and k 2 6 (b+c)b , 2+ 3 0.2 < θ2 6 1, ν := −2 cb > 0, then ³ c´ ku(x) 6 α0 − η(x) 6 νη(x), x ∈ IR, (14) b and

0 > ku0 (x) > νη 0 (x), x ∈ (0, ∞), where

½ α0 := sup

x∈IR

2kbη + 2kcη − cuη 2bu + buη

(15)

¾ < ∞.

(16) ¥

The fact that (u(0), η(0)) lies on F is central to the analysis. Not only can we use it to estimate the speed k of a solitary wave with given (u(0), η(0)),(cf. (6)) but also to estimate the slope of F at that point: Lemma 3.3. If (u, η) satisfies (9), then (u(0), η(0)) ∈ F at a point, where the slope du/dη of F satisfies the inequalities du c 6 6 0. kb dη

(17) ¥

7

We finally state the uniqueness theorem which can be proved with the aid of the previous Lemmata in analogy with Theorem 3.2 of [12]: 3c+b 2 Theorem 3.1. If u(0) k 6 3c and k 6 solution of the problem (9) is unique.

Remark: By (10), if k 6

−6c b−3c , 2

account the assumption that k 6

then c2 (b+c)b

√ 2+ 0.2 c2 (b+c)b , 3

u(0) k

6

3c+b 3c .

< θ2 6 1, then the ¥ Hence, taking into

it follows, from Theorem 3.1 that √

a sufficient condition for uniqueness is that θ2 ∈ ( 2+ 3 0.2 , 1] and that ¾ ½ n o q 2(3θ 2 −2) 12(3θ 2 −2) −6c c2 √ k 6 min b−3c , (b+c)b = min 21θ2 −13 , =: kmax (θ). 2 2 (3−3θ )(3θ −1)

If θ2 = 1, we recover the kmax = 3/2 of [12]. 4. Numerical Studies of the Stability of Solitary Waves The question of stability of the solitary wave solutions of two-way propagation models, such as the Boussinesq systems is an important problem. No rigorous results exist yet, for example, about the nonlinear orbital stability of the solitary waves. In particular, the classical approach, cf. e.g. [1], [8], fails in the case of the Bona - Smith system (1) because the solitary wave, a stationary point of an appropriate constrained variational problem for a functional associated with the Hamiltonian structure of (1), turns out to be a saddle point of infinite index. It is of interest then to study numerically the effects of small and large perturbations on the evolution of solitary waves to provide clues and motivation for the analysis. Such a study has been carried out in [6], where initial solitary wave profiles have been perturbed in a wide collection of directions and magnitudes; the periodic initial-value problem for (1) has then been integrated numerically using a standard Galerkin finite element scheme with cubic splines in space and the classical explicit Runge-Kutta method for the temporal discretization. The results indicate that the solitary waves are, apparently, asymptotically stable. For example, if a solitary wave of the form (7)–(8) for θ2 = 9/11, x0 = 100 is perturbed so that η0 (x) = rηs (x, 0) with r = 1.1, there results one η-solitary wave of amplitude 1.06110 (note that the amplitude of the unperturbed initial profile ηs (x, 0) was equal to one), that travels to the right with a speed k = 1.4673 plus right- and left-travelling oscillatory dispersive tails of small, decaying amplitude. A magnification of the solution at t = 100 appears in Fig. 2. The solitary wave is the big pulse located near x = 50 and followed by the part of the dispersive tail

8

that is travelling to the right. The other part of the tail has wrapped itself around x = 150 by periodicity and is travelling to the left. 0.03 0.02 0.01 0 -0.01 -0.02 -150

Figure 2.

-100

-50

0

Evolution of a perturbed η-solitary wave,

50

θ2

100

150

= 9/11, r = 1.1, t = 100.

Larger initial perturbations may lead to more than one solitary waves plus dispersive tails. For example, if r = 1.8, in addition the main righttravelling solitary wave (of η-amplitude about 1.48043 and speed 1.6242), a second solitary wave (of η-amplitude about 0.251 and speed k = −1.12) emerges quite early at the head of the left-travelling dispersive tail. (By t = 80, cf. Fig. 3, the left-travelling wave train has wrapped itself around the right-hand boundary due to periodicity.) 1.6 1.4 1.2 1 0.8 0.6 0.4 0.2 0 -0.2 -150

Figure 3.

-100

-50

0

Evolution of a perturbed η-solitary wave,

50

θ2

100

150

= 9/11, r = 1.8, t = 80.

It is worthwhile to note that large perturbations, for example of the initial u-solitary wave profile, lead to apparent blow-up instabilities in finite time after η becomes less than −1 at some point (x∗ , t∗ ). In an example where we took u0 (x) = 3.8us (x, 0), where us , ηs were given by (7)–(8) and θ2 = 9/11, x0 = 100, η developed a negative excursion that became less than −1 at about t = 3. Subsequently, the solution apparently developed a singularity and blew up.

9

Acknowledgments This work was supported in part by the Institute of Applied and Computational Mathematics of FO.R.T.H. and in part by a grant of the Research Committee of the University of Athens. References 1.

T. B. Benjamin, The stability of solitary waves, Proc. Roy. Soc. Lond. A 328(1972), 153–183. 2. J. L. Bona, M. Chen, and J. -C. Saut, Boussinesq equations and other systems for small-amplitude long waves in nonlinear dispersive media. Pt I: Derivation and the linear theory, J. Nonlin. Science 12(2002), 283–318. 3. J. L. Bona and R. Smith, A model for the two-way propagation of water waves in a channel, Math. Proc. Camb. Phil. Soc. 79(1976), 167–182. 4. M. Chen, Solitary-wave and multi pulsed traveling-wave solutions of Boussinesq systems, Applic. Analysis 75(2000), 213–240. 5. M. Chen, Exact traveling-wave solutions to bi-directional wave equations, Int. J. Theor. Phys. 37(1998), 1547–1567. 6. V.A. Dougalis, A. Duran, M.A. Lopez-Marcos, and D.E. Mitsotakis, A numerical study of the stability of solitary waves of the Bona-Smith system, (to appear). 7. B. Gidas, W. -M. Ni, and L. Nirenberg, Symmetry and related properties via the maximum principle, Commun. Math. Phys. 68(1979), 209–243. 8. M. Grillakis, J. Shatah, and W.A. Strauss, Stability of solitary waves in the presence of symmetry I, J. Funct. Anal. 74(1987), 170–197. 9. D. E. Mitsotakis, Solitary waves for the Bona-Smith system: Existence, uniqueness and stability, (in Greek) M.Sc. Thesis, Mathematics Dept., Univ. of Athens, 2003. 10. J. F. Toland, Solitary wave solutions for a model of the two - way propagation of water waves in a channel, Math. Proc. Camb. Phil. Soc. 90(1981), 343–360. 11. J. F. Toland, Existence of symmetric homoclinic orbits for systems of EulerLagrange equations, A.M.S. Proceedings of Symposia in Pure Mathematics, v.45, Pt. 2 (1986), 447–459. 12. J. F. Toland, Uniqueness and a priori bounds for certain homoclinic orbits of a Boussinesq system modelling solitary water waves, Commun. Math. Phys. 94(1984), 239–254.