Research Article | Open Access
Lynnyngs Kelly Arruda, "On the Instability of a Class of Periodic Travelling Wave Solutions of the Modified Boussinesq Equation", International Journal of Mathematics and Mathematical Sciences, vol. 2010, Article ID 637497, 21 pages, 2010. https://doi.org/10.1155/2010/637497
On the Instability of a Class of Periodic Travelling Wave Solutions of the Modified Boussinesq Equation
This paper is concerned with instability of periodic travelling wave solutions of the modified Boussinesq equation. Periodic travelling wave solutions with a fixed fundamental period L will be constructed by using Jacobi's elliptic functions. It will be shown that these solutions, called dnoidal waves, are nonlinearly unstable in the energy space for a range of their speeds of propagation and periods.
The original Boussinesq equations are among the classical models for the propagation of small amplitude, planar long waves on the surface of water [1, 2]. These equations possess special travelling wave solutions known as Scott Russel’s solitary waves or solitons [3, 4], cnoidal waves , and dnoidal waves (, Section 3 below). The cnoidal and dnoidal wave solutions are periodic travelling waves written in terms of the Jacobi elliptic functions.
Our purpose is to investigate the nonlinear stability of periodic travelling wave solutions of the modified Boussinesq equation The above equation (1.1), has the following equivalent form as a Hamiltonian system for , . Here subscripts and denote partial differentiation with respect to and .
The above equation conserves energy, namely, the integral does not depend on the time . Another conservation law is the momentum which turns out to be a relevant quantity in the investigation of stability properties of travelling waves.
To make precise the notion of stability we use, let be the translation by , for and let be an -periodic travelling wave solution to system (1.2), where , , is the period of and , and is the wave’s speed of propagation. If we define the -orbit to be the set , is called orbitally stable if profiles near its orbit remain near the orbit for as long as it exists.
So, we have the following definition. Let be a Hilbert space.
Definition 1.1 (Orbital Stability). Let be an -periodic travelling wave solution to system (1.2). We say that the orbit is stable in the -sense by the flow of system (1.2) if for each there exists such that if and then the solution of (1.2) with satisfies, for all for which exists, Otherwise, we say that is -unstable.
Here, . (The choice of norm in (1.5) is dictated by the form of the Hessian or “linearized Hamiltonian” and varies from problem to problem.)
Inserting the -periodic travelling wave solution in (1.2) leads to the system where connotes and . Integrating the latter system, we obtain the nonlinear system where are integration constants, which will be considered equal to zero here. Then, we obtain Next observe that relation (1.8) characterizes as a critical point of subject to the constraint In order to prove instability for , we will examine the relation between the concavity properties of the function and the properties of the functional near the critical point under the constraint .
Bona and Sachs in  proved that the well-known solitary waves of the generalized Boussinesq equation are stable in the norm for speeds such that if in (1.9) is a convex function of . The aim of this paper is to prove that the solutions given by Theorem 3.2 below are unstable if is concave. The proof follows the main ideas of Liu  (see also Bona et al. in ). Differently from the solitary wave solutions case, we do not know explicit periodic travelling wave solutions in the -variable for the system (1.10) for every . For this reason, we will treat here only the case . Stability of dnoidal waves for this case is also treated by the author in a forthcoming paper . Regarding the classical case , in  the author proves nonlinear stability properties of a class of -periodic travelling wave solutions, called cnoidal waves, in the energy space , by periodic disturbances with period .
In this paper, we first show the existence of a smooth curve of dnoidal wave solutions to system (1.2), with a fixed period (Theorem 3.2 below). Then, a proof of orbital instability of these solutions is established in for a certain range of their speeds of propagation and periods, based on a modification of the general procedure of . More precisely, our main result regarding stability of the dnoidal waves , given by Theorem 3.2 below, is the following.
Theorem 1.2 (Instability Theorem). Let and . Then the orbit is -unstable with respect to the flow of the modified Boussinesq equation, provided and .
The plan of this paper is as follows. A discussion of the evolution equation (1.1) and its natural invariants is given in Section 2. In Section 3 we introduce a smooth family on the parameter , of positive dnoidal wave solutions to system (1.2), with a fixed period (Theorem 3.2 below) and in Section 4 we present a complete study of the spectrum of the operator . The existence of the smooth curve will allow us to differentiate the function . Then, in section 5, we prove that is indeed concave, for a certain range of speeds and periods of , which will imply our result. In Section 6 the Lyapunov functional [8, 9] is constructed and the instability result is proved. In Appendix, we give a review of those results about Jacobian elliptic functions which we use throughout the paper.
We remark that orbital instability of is established with respect to perturbations of periodic functions of the same period in .
The following notation will be used:
2. The Evolution Equation
The next lemma is the periodic version of a particular case of [4, Lemma ].
Lemma 2.1. Let . Then there exist and a uniquely weak solution of (1.2) with
Proof. In order to obtain the existence of weak solutions for the system (1.2), we consider the approximate problem
with and in , where
and is the infinitesimal generator of a group of unitary operators in and . Since , the map is locally Lipschitz on . But then for all , there exists a such that the initial value problem (2.1) has a unique solution . Moreover, if , then
by the semigroup theory . By (2.1), we estimate on
where we used in the first equality above that .
Consider now , which is a continuous, positive and increasing function on . Then by Gronwall’s inequality, it follows that
We compare with the maximal solution of the scalar Cauchy problem It follows that
Let . Then is defined on for all . Moreover, on , where is a constant independent of , since by (2.5), (2.8), and the fact that is bounded on , we have the following inequality on :
Finally, from (2.9) and standard weak limit arguments, we have the existence of a unique solution .
The proof is elementary.
3. Existence of a Smooth Curve of Dnoidal Wave Solutions with a Fixed Period for the System (1.2)
This section is devoted to establish the existence of a smooth curve of periodic travelling wave solutions for the system (1.2), which are solutions of the form
Substituting (3.1) in (1.2) leads to the system where denotes and . Integrating (3.2), we obtain the nonlinear system where are integration constants, which will be considered equal to zero here. Then, must satisfy where will be considered positive.
Next, we show how to construct a smooth curve of solutions for (3.4) with a fixed fundamental period , and depending on the parameter . In order to do this, we first observe from (3.4) that satisfies the first-order equation where is an integration constant and , , , are the real zeros of the polynomial , which satisfy the relations Moreover, we assume without lost of generality that and we obtain from (3.5) that . By defining and , (3.5) becomes . We also impose the crest of the wave to be at , that is, . Now, we define a further variable via the relation and so we get that Then we obtain for that Therefore, from the definition of the Jacobian elliptic function (see in the appendix or in Byrd and Friedman ), we can write the last equality as and hence Returning to the initial variable, we obtain the called dnoidal wave solution associated to (3.4), with Next, dn has fundamental period , , where represents the complete elliptic integral of the first kind (see appendix); it follows that the dnoidal wave in (3.7) has fundamental period, , given by Now, we show that . First, we express as a function of and . In fact, for every , there is a unique satisfying the first relation in (3.6), namely, . So, from (3.9) we obtain
Then, by fixing , we have that as and as . So, since the mapping is strictly decreasing (see proof of Proposition 3.1), it follows that .
Now, we obtain a dnoidal wave solution with period . For , there is a unique such that . So, for such that , the dnoidal wave has a fundamental period and satisfies (3.4) with .
By the above analysis the dnoidal wave in (3.7) is completely determined by and and will be denoted by or .
Proposition 3.1. Let be arbitrary but fixed. Consider and the unique such that . Then, (1)there exist an interval around , an interval around , and a unique smooth function such that and where , , and is defined by (3.10); (2)the positive dnoidal wave solution in (3.7), , determined by , , has fundamental period and satisfies (3.4). Moreover, the mapping is a smooth function; (3) can be chosen as ; (4)the mapping is strictly decreasing.
Proof. From this result we conclude the following existence theorem.
Theorem 3.2. Let . Then there exists a smooth curve of dnoidal wave solutions for the system (1.2) in , which satisfy the system (3.3) with integration constants ; this curve is given, for , by
Moreover, where the smooth function is given by Proposition 3.1 and by (3.10).
Remark 3.3. is in as soon as in . This follows from the equation and a bootstrap argument.
4. Spectral Analysis
In this section, we study the spectral properties associated to the linear operator determined by the periodic solutions found in Theorem 3.2. We compute the Hessian operator by calculating the associated quadratic form, which is denoted by . By definition, is the coefficient of in and so is given by Note that is the sum of the quadratic form associated to the operator and the nonnegative term . From (3.2) for the dnoidal wave , it follows that and satisfy . To see that this is the only eigenfunction corresponding to the eigenvalue zero and the other expected properties of the operator , we will first consider the following result about the periodic eigenvalue problem: where is given by Proposition 3.1.
Theorem 4.1. Let be the linear operator defined on by (4.4). Then the first three eigenvalues , , and of are simple, and satisfy , and is the eigenfunction of . Moreover, the rest of the spectrum consists of a discrete set of eigenvalues which are double.
To prove that the kernel of is spanned by , consider the quadratic form as the pairing of against in the duality, where is the unbounded operator applied to . Then implies Now, from the properties of the operator established in Theorem 4.1, it follows that and , where .
To show that there is a single, simple, negative eigenvalue, consider defined in (4.3) above. By Theorem 4.1, the operator has exactly one negative eigenvalue which is simple, say , with associated eigenfunction . Thus, achieves a negative value and so does . In fact, considering , we have Denoting by the lowest eigenvalue of , we will show that the next eigenvalue is , which is known to be simple, and consequently is in fact strictly positive. These results are proved using the (min-max) Rayley-Ritz characterization of eigenvalues (see [14, 15]), namely, Choosing , , we obtain the lower estimate The right-hand side of (4.8) is nonnegative on the subspace since by Theorem 4.1. Thus, and, from earlier considerations, is simple and .
The above analysis can be summarized in the form of the following theorem.
Theorem 4.2. Let be the linear operator defined on by (4.1). Then the first two eigenvalues and of are simple and satisfy ; , with and being the eigenfunctions of and , respectively. Moreover, the rest of the spectrum consists of a discrete set of eigenvalues and the mapping is continuous with values in .
5. Concavity of
Lemma 5.1. Let and . Then the function is concave, provided and .
Remark 5.2. Relation (1.8) implies that is equivalent to the condition
Proof of Lemma 5.1. Note that
Indeed, we observe from (3.7), (3.8), and (3.11) that
where we used the fact that the Jacobi elliptic function has fundamental period and is an even function. Now, by using that and , it follows from (5.4) that
Now, Proposition 3.1 and Theorem 3.2 imply that the map is strictly decreasing and from (3.10), with , we have that
Thus, since is strictly increasing (see Appendix), the claim (5.3) follows from (5.5) and (5.6).
So, from (5.2), (5.3), and (5.5), we get Now, considering the function defined by (2.12) in  and using (5.6), we obtain hence
From (5.7), (5.9), and using that , we obtain or equivalently, where .
Now, given that , we rewrite the coefficient of in (5.10) as Also, the coefficient of can be rewritten as Thus,
We remark that we can write as a function of complete elliptic integrals. In fact, by integrating (3.4) from to , we obtain which is well defined, since the solution is positive.
Now, using (3.7), the expression in , and the fact that (see Appendix), we obtain Similarly using (3.7), the expression in , and the special values , and , (see Appendix), it follows that Substituting (5.14) and (5.15) in (5.13), we deduce that Using (5.16) and , the numerator of (5.12) will be positive if and only if
Remark 5.3. since the functions and are strictly increasing (see Appendix).Claim 1. Proof. Indeed, denoting by and by , we use Hospital’s rule to find the limit (5.17). Specifically, we show using (A.3) that which implies our claim.
Note that, by , we have that Moreover, , since . In, addition we get , since the function has the following properties: and . We conclude that the function is strictly positive on . Now, continuity plus (5.17) and (5.19) implies that satisfies .
This concludes the lemma.
Consider the function defined by (1.9). We now examine the relation between concavity properties of and the properties of the functional near the critical point subject to the constraint .
Theorem 6.1. Let be fixed. If , then there exists a curve which satisfies , , and on which has a strict local maximum at .
Proof. We follow the ideas of [3, 8, 9]. Let be the unique, negative eigenfunction of . Define , for near , where satisfies and . The function can be defined by the implicit function theorem, since where with , and is the unique negative eigenvalue of and , . Thus It is easy to see that where In fact, by some calculations, we have that so that in view of the fact that .
Lemma 6.2. There exist and a unique map , such that, for any and , where is the “tube”
Definition 6.3. For , define by the formula where .
By Lemma 6.2, may also be expressed as
Lemma 6.4. is a function from into , commutes with translations, , and for any .
Lemma 6.5. There exists a function which is invariant under translations, such that if with and not being a translation of , we have
Lemma 6.6. The curve constructed in Theorem 6.1 satisfies for , and changes sign as passes through , with .
6.1. Proof of Theorem 1.2
First, consider with . Let be given small enough. By Lemma 6.6, we can choose arbitrarily close to such that , e . To prove the instability of , it suffices to show that there are some elements which are close to but for which the solution