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On travelling waves in a suspension bridge model as the wave speed goes to zero✩ A.C. Lazer a , P.J. McKenna b,c,∗ a

Department of Mathematics, University of Miami, Coral Gables, FL, United States

b

Department of Mathematics, University of Connecticut, Storrs, CT 06268, United States

c

KUFPM, Dahran, Saudi Arabia

article

info

Article history: Received 8 March 2011 Accepted 14 March 2011 Communicated by Ravi Agarwal

abstract Travelling waves exist in nonlinearly supported beams. It is shown that as the wave speed goes to zero for a special type of nonlinearity, the amplitude of the waves must go to infinity. © 2011 Elsevier Ltd. All rights reserved.

Keywords: Nonlinear beam Suspension bridgesep Travelling waves

In the past twenty five years, there has been considerable research into nonlinear partial differential equations which are conceived to model various aspects of oscillations of suspension bridges. Typically these equations are of the type utt + uxxxx + f (u) = 1.

(1)

This equation is used to model a suspension bridge as a vibrating beam being supported by cables which have a nonlinear response to loading, namely be resisting expansion but not compression, and a constant weight per unit length due to gravity. The unknown u(x, t ) measures deflection from the unloaded state and is therefore applicable to vertical oscillations. Of course, suspension bridges have a variety of different types of oscillation, including vertical but more famously, as in the Tacoma Narrows case, [1], also travelling waves, [1]. The first papers, [2,3], usually took f (u) = u+ in (1). Thus the cables were seen as obeying a linear Hooke’s law under expansion but not resisting compression from the unloaded state. Most of the papers, [2,4] were concerned with periodic solutions of (1) and so the variable x was taken on an interval of length L. Then with a simplifying assumption, it is possible to reduce Eq. (1) to an ordinary differential equation and study periodic solutions for this, sometimes with small added forcing and damping. The periodic solutions of the full partial differential equation were also studied in [2]. This leads to the study of periodic solutions of utt + uxxxx + bu+ = 1 + ϵ h(x, t )

in [0, π] × R

u(0, t ) = u(L, t ) = uxx (0, t ) = uxx (L, t ) = 0.

(2)

The other situation which is the object of much study is travelling wave solutions of (1). This study was initiated in [3]. Now the problem becomes, (after some simplifying normalisations), utt + uxxxx + u+ = 1 ✩ This paper is dedicated to the memory of our friend and colleague Wolfgang Walter, (1927–2010).

(3)

∗ Corresponding author at: Department of Mathematics, University of Connecticut, Storrs, Ct. 06268, United States. Tel.: +1 203 486 3989; fax: +1 860 486 4238. E-mail addresses: [email protected], [email protected], [email protected] (P.J. McKenna). 0362-546X/$ – see front matter © 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.na.2011.03.024

A.C. Lazer, P.J. McKenna / Nonlinear Analysis 74 (2011) 3998–4001

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where now x is defined on the entire real line. This equation has an obvious equilibrium at u ≡ 1 and to search for travelling waves, we look for solutions Eq. of (3) of the form u(x, t ) = 1 + y(x − ct ) where y decays to zero exponentially as |x| → ∞. Thus we are concerned with finding homoclinic solutions of the nonlinear ordinary differential equation y′′′′ + c 2 y′′ + (1 + y)+ − 1 = 0.

(4) ′′′′

2 ′′

In [3], the approach taken was to write analytic expressions for the solutions of y + c y + y = 0 for y ≥ −1 and y′′′′ + c 2 y′′ − 1 = 0 for y ≤ −1. Having found these two expressions, the task of finding solutions of (4) became the task of ensuring continuity of the solution y and its first three derivatives whenever y = −1. This leads to four transcendental equations, which in turn are simplified to finding the zeros of a single function of two variables. The zero set of this function √ is then found numerically for some values of c in the interval 0 < c < 2. There are some problems with this approach. First, existence was not proved rigorously since all calculations were approximate. Nonetheless, the paper led to some obvious conjectures. First it seemed that the number of solutions could be quite large, especially as c → 0. The second observation made in [3] was that as c → 0, the L∞ -norm of the solutions seemed to go to +∞. Several examples of this were shown in [3]. Furthermore, the method of ‘‘finding’’ these traveling wave solutions was heavily dependent on the analytic form of the nonlinearity of (4). Thus, the publication of this first paper left open several interesting questions. First, could one give a general rigorous proof for the existence of homoclinic orbits of (4) for all c in the interval √ 0 < c < 2? Second, could one verify that as c → 0, the L∞ -norm of the solutions seemed to go to +∞, as was indicated by the computations? Third, could one prove existence (and multiplicity) for solutions of a more general nonlinearity with the same basic shape as that of (4)? And fourth, did these travelling waves have interesting stability, instability, or interaction (soliton) properties? This fourth problem became important with the publication of [5,6], where the answer to the first question was given. In this paper, the mountain pass theorem, with concentrated compactness, was used to show existence for all c. More recently, a different variational approach was used in [7]. Furthermore, many computational experiments were performed seeming to show that for the piecewise nonlinearity, the travelling waves had some interesting properties, although the non-smoothness of the nonlinearity made the numerical experiments suspect. Thus, a smoothed version of f (y) = (1 + y)+ − 1 was introduced and homoclinic orbits of y′′′′ + c 2 y′′ + ey − 1 = 0

(5)

were studied. Note that this nonlinearity has the shape of the earlier one, having slope 1 at zero, going to +∞ as y → +∞ and going to −1 as y → −∞. This new ordinary differential equation, and the corresponding partial differential equation apparently had many solutions which could be calculated by the mountain pass algorithm. Furthermore, they were shown to have interesting stability, soliton, and fission properties. However, the basic method of proof for the existence of the solutions via the mountain pass theorem did not seem to work. This left additional open problems. Could one prove the existence of at least one solution, or preferably more than one solution? Could one investigate further the stability properties of these waves in one space dimension or even in two dimensions, with the corresponding equation for a supported plate

∆2 u + c 2 uxx + eu − 1 = 0

(6)

either rigorously or numerically? While as of yet there is no information on rigorous stability of the traveling waves, there has been some progress on the numerical front, and also on the rigorous proof of existence. In particular, in [8], existence √ of solutions of (5) was shown for almost all c in n, the interval 0 < c < 2. Separately, in [9], the existence of thirty six solutions was rigorously proved for one particular value of c. More recently, in [10], existence has been established for all c in the partial interval (0, 0.7427). In this paper, we give an analytic proof of the original speculation of [3], namely that for Eq. (4), then as c → 0, the L∞ -norm of the solutions seemed to go to +∞. We shall conclude with some other open problems. 1. The main result We begin with a non-existence theorem about what happens when c = 0. Theorem 1. The solution u ≡ 0 is the only solution of the equation y′′′′ + (1 + y)+ − 1 = 0 such that ‖y‖L∞ is bounded.

(7)

4000

A.C. Lazer, P.J. McKenna / Nonlinear Analysis 74 (2011) 3998–4001

Proof. Let g (u) = (u + 1)+ − 1. Then g (u) satisfies ug (u) ≥ k|u| and ug (u) ≥ k(u+ )2 . Let u be a solution of (7) bounded by M. Then by the mean value theorem, there there exists a point t such that

|u′ (t )| ≤ 2M on any interval of length one. By the same argument, |u′′ | ≤ 4M at some point of every interval of length three. Similarly, |u′′′ | ≤ 8M at some point of any interval of length seven. Since u(4) is bounded everywhere on (−∞, ∞), it follows that all four derivatives are bounded each on (−∞, ∞). Let F (t ) = u′′′ u − u′′ u′ . Note that F is bounded on R. An easy calculation gives that F ′ = u(4) u + u′′′ u′ − (u′′ )2 − u′′′ u′ == −ug (u) − (u′′ )2 . Thus −F (t ) ≥ k|u| + (u′′ )2 . Therefore, F (−t ) − F (t ) = −

∫

t

F ′ (s)ds ≥ −t

∫

t

(k|u| + (u′′ )2 )ds. −t

t

However, for some M , |F (t )| ≤ M since u and hence F are bounded on R. In turn, this implies that −t (k|u| + (u′′ )2 )dt ≤ 2M. This implies that u → 0, u′′ → 0 as |t | → ∞. Therefore F (t ) → 0 as |t | → ∞. In turn, this implies that u ≡ 0. Theorem 2. As c → 0, all solutions of y′′′′ + c 2 y′′ + (1 + y)+ − 1 = 0 must go to +∞ in the L∞ norm. Proof. Suppose not, then there exists a sequence cn ↓ 0, with corresponding solutions yn such that ‖yn ‖∞ ≤ M for some M. Since the nonlinearity (1 + y)+ − 1 is linear for y ≥ −1 and the linear equation y′′′′ + c 2 y′′ + y = 0 admits no bounded solutions for c close to zero, it follows that there exists points τn such that yn (τn ) ≤ −1 for all n. Without loss of generality, replace yn (t ) by yn (t + τn ). By passing to subsequences, if necessary, we can assume (since all derivatives (k) are bounded) that yn (0) → αk for k = 1, 2, 3, 4. Thus the solutions yn (t ) converge uniformly on compact subsets to a bounded solution of y′′′′ + (1 + y)+ − 1 = 0 where y(0) ≤ −1 and y(k) (0) = αk for k = 1, 2, 3, 4. This contradicts the conclusion of Theorem 1, so as c → 0, all solutions must become unbounded. 2. Concluding remarks and open questions The most obvious question is whether the same conclusions can be reached for the ‘‘more interesting’’ nonlinearity y = ey − 1 since engineers, (for example [11]) are interested in smoothed versions rather than the piecewise linear one. Mathematically, we would also like a more general class of nonlinearity f . We would need some way to show that as cn ↓ 0, there exists a sequence of solutions yn,cn such that sup{yn,cn } remains bounded away from zero. In the case of the piecewise linear nonlinearity, this was obvious. In the exponential case or other more general case, we would need something about the invertibility of the linearised operator near the zero solution in some reasonable space since y ≡ 0 is already a solution and if the linearised operator near 0 is nondegenerate, we would be done. The trouble is that the linearised operator is not invertible because of translation. If we knew that all solutions were symmetric about about zero, we could remove this problem by working in a symmetric subspace of H 2 (−∞, ∞) (see for example [9]) in which case the solution of the linearised problem around y ≡ 0 is nondegenerate. However, the symmetry of these solutions about t = 0 (after a suitable translation) remains open. Although many computational solutions have been obtained (see, for example, [9,12,5]) no unsymmetric solutions have been observed. A second natural question is exactly how these solutions behave as cn ↓ 0 and how many there are. A natural conjecture, based on the numerical evidence, is that the number of solutions goes to infinity as cn ↓ 0.

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