Convergence Of Solitary-Wave Solutions In A Perturbed Bi-Hamiltonian Dynamical System. I. Compactons And Peakons

We investigate how the non-analytic solitary wave solutions -- peakons and 
compactons -- of an integrable bi-Hamiltonian system arising in fluid mechanics, can be 
recovered as limits of classical solitary wave solutions forming analytic homoclinic orbits for 
the reduced dynamical system. This phenomenon is examined to understand the important 
effect of linear dispersion terms on the analyticity of such homoclinic orbits.

1. Introduction.Classically, the solitary wave solutions of nonlinear evolution equations are determined by analytic formulae (typically a sech2 function or variants thereof) and serve as prototypical solutions that model physical localized waves.In the case of integrable systems, the solitary waves interact cleanly, and are known as solitons.For many examples, localized initial data ultimately breaks up into a nite collection of solitary wave solutions; this fact has been proved analytically for certain integrable equations such as the Korteweg-deVries equation, 2], and is observed numerically in many others.More recently, the appearance of non-analytic solitary wave solutions to new classes of nonlinear wave equations, including peakons, 6], 13], which have a corner at their crest, cuspons, 24], having a cusped crest, and, compactons, 17], 18], 20], which have compact support, has vastly increased the menagerie of solutions appearing in model equations, both integrable and non-integrable.The distinguishing feature of the systems admitting non-analytic solitary wave solutions is that, in contrast to the classical nonlinear wave equations, they all include a nonlinear dispersion term, meaning that the highest order derivatives (characterizing the dispersion relation) do not occur linearly in the system, but are typically multiplied by a function of the dependent variable.
The rst and most important of the nonlinearly dispersive, integrable equations is the equation u t + u xxt = u x + u xxx + 3 uu x + (uu xxx + 2u x u xx ): (1.1) Here , , and are real constants and u(x; t) is the unknown function depending on the temporal variable t and the spatial variable x.This equation contains both linear dispersion terms u xxt , u xxx , and the nonlinear dispersion terms uu xxx .Equation (1.1) can, in certain parameter regimes, be regarded an an integrable perturbation of the wellknown BBM (or regularized long wave) equation u t + u x + uu x ?u xxt = 0; (1.2) which was originally proposed, 4], as an alternative to the celebrated Korteweg-deVries (KdV) equation u t + u x + uu x + u xxx = 0; (1.3) in the modeling of dispersive nonlinear wave phenomena.Although the integrability of the KdV equation makes it a more mathematically signi cant equation (see 2]), the BBM equation has better analytical properties, including the more desirable linear dispersion relation for uid modeling, 4].Equation (1.1) rst appears (albeit with a slight error in the coe cients) in the work of Fuchssteiner (ref. 8], Equation (5.3)).Camassa and Holm,6], derived (1.1), for certain values of the coe cients including < 0, as a model for water waves, and established an associated linear scattering problem.They began a systematic study of the solutions of (1.1), discovering that its soliton solutions are only piecewise analytic, having a corner at their crest, and hence named them peakons.Although not classical solutions, peakons do form weak solutions of (1.1).Like classical solitons, there exist multi-peakon solutions of (1.1) with cleanly interacting peakons; see 3] for a detailed analysis of their behavior.At the same time, Rosenau,18], discovered a wide variety of nonlinear wave equations with nonlinear dispersion that admit compactly supported solitary wave solutions | epitomized by the (presumably) nonintegrable family of equations u t (u m ) x + (u n ) xxx = 0; (1.4) depending on positive integers m; n.If n 2, (1.4) admits a one-parameter family of compactons, the parameter being the wave speed (which also governs its amplitude).Inspired by this discovery, Rosenau proposed the alternative form of Equation (1.1), with > 0, and claimed that it provided an example of an integrable system supporting compacton solutions.However, very few of Rosenau's proposed compactons are actually solutions of (1.1) in the weak sense | indeed we shall prove that there is at most a single wave speed for which (1.1) with > 0 admits a compacton solution vanishing at in nity.An additional di culty with this equation is that, for > 0, the dispersion relation for (1.1) has singularities which raises questions about it being a well-posed initial value problem.The nonlinear wave equation (1.1) is just one of a wide variety of examples of \dual Hamiltonian systems" which can be constructed from classical soliton equations such as the KdV equation by either a Lagrange transformation, 19], or, more generally, a rearrangement of the operators appearing in their bi-(or, rather, multi-) Hamiltonian structure; the basic method appears in 9], and has been extensively developed in 7], 16], to which we refer the reader for many additional interesting examples.We remark that certain versions of the Harry Dym equation derived by this dualization procedure do admit parametrized families of compactons and are integrable (at least in the sense that they appear in a Hamiltonian hierarchy).
Although one can characterize the compactons and peakons as weak solutions, we remark that they are solutions in a considerably stronger (although not quite classical) sense than the more traditionally studied weak solutions such as shock waves.As a matter of fact, they are piecewise analytic, satisfying the equation in a classical sense away from singularities, and, moreover, each term (or certain combinations of terms) in the equation have well de ned limits at the singularities.These facts will be demonstrated in Sections 6 and 7 in Part II of this paper.Indeed, one does not require any entropy condition to prescribe the type of singularity.Thus, we propose an apparently new and potentially useful de nition of such \pseudo-classical" solutions that will handle a wide variety of such non-analytic solitary wave solutions, and distinguish them from shock waves.
For classical, linearly dispersive systems, the characterization of solitary waves and more complicated solutions by their analyticity properties has been the focus of a signi cant amount of study.Kruskal,12], proposed analyzing the interactive properties of solitary wave solutions by the behavior of their poles in the complex plane.The Painlev e test for integrability of nonlinear systems, 1], 22], 25], is based on the analyticity of their solutions.The convergence of the general Painlev e series expansions has been studied in 11].Recently Bona and Li, 5], showed that for various types of linearly dispersive systems, including equations of both KdV and BBM type, all weak solitary wave solutions which are essentially bounded and decay to zero at in nity are necessarily classical solutions, and can be analytically extended to a horizontal strip in the complex plane containing the real axis.In contrast, we will demonstrate that the system (1.1) not only has solitary wave solutions that are restrictions of analytic functions de ned on a horizontal strip, but also admits both compactons, whose second order derivatives are discontinuous, and peakons, whose rst order derivatives have a discontinuity, as weak (or, rather, pseudo-classical) solutions.The existence of such types of non-analytic solutions requires that the linear dispersion term vanishes for certain function values, and these are precisely the values at which the discontinuities of the solution appear.This fact indicates the important role played by the linear dispersion terms in the formation of analytic travelling wave solutions, and the signi cant in uence of nonlinear terms on the behavior of these solutions.
The appearance of non-analytic solutions thus draws our attention to a more detailed understanding of the e ects of both linear and nonlinear dispersion terms on travelling wave solutions, especially, on solitary wave solutions.In this paper, we shall discuss how such non-analytic solitary wave solutions can appear as the limits of classical, analytic solitary wave solutions.(This observation is not as paradoxical as it might initially seem | a classical instance of such loss of analyticity occurs in the convergence of Fourier series.)Such limits can be e ected in two di erent, but essentially equivalent ways.First one can add to the equation a small linear dispersion term, having the e ect of forcing analyticity of the perturbed solitary wave solution, and then allowing the coe cient of this additional term to vanish.In (1.1), the coe cient has this e ect, provided we compensate by setting = ~ = to leave the nonlinearly dispersive term uu xxx intact.This approach is, in part, motivated by the convergence properties of the KdV equation as the dispersion term (meaning the coe cient of u xxx ) goes to zero; the convergence of classical solutions to the KdV equation to non-analytic shock wave solutions of the resulting dispersionless Burgers' equation u t + uu x = 0 was analyzed in great detail by Lax, Levermore and Venakides, 14], 15], 23].Of course, our situation is analytically simpler since the limiting equation does not have shocks, and, besides, we are only attempting to analyze solitary wave solutions.Alternatively, one can replace the vanishing condition at in nity by the condition u ! a as jxj ! 1, meaning that the wave appears as a disturbance on a uid of uniform depth a.For most values of a, this has the e ect of eliminating the e ect of the nonlinearly dispersive terms, and again one can investigate how the associated analytic solitary wave solutions lose their analyticity as the undisturbed depth a !0. Of course, these two approaches are closely related | one can replace u by û = u?a to eliminate the non-zero asymptotic depth; the resulting equation will then include an additional linearly dispersive term depending on the small parameter a (as well as additional nonlinear terms).
In this paper we investigate both strategies for (1.1) | here the transformation û = u ? a merely rede nes the parameters ; ; ; appropriately.Our main result is that, in all cases, analytic solitons converge to non-analytic peakons and compactons provided the latter are weak (i.e.pseudo-classical) solutions to the system.Thus the convergence of analytic solitary wave solutions under vanishing linear dispersion is to pick out those nonanalytic solitary wave solutions which are \genuine" in the sense that they are weak, or pseudo-classical, solutions.We anticipate that this will form a rather general convergence phenomena, applicable to both integrable and nonintegrable systems, including (1.4), alike.
Our analysis breaks into three parts.First, methods from the theory of dynamical systems | in particular center manifold theory | will be employed to produce a preliminary analysis of the ordinary di erential equations describing travelling wave solutions to Equation (1.1).This allows us to determine the precise parameter regimes for which (1.1) admits solitary wave solutions, which can always be characterized as the limit of periodic travelling wave solutions, as well as peakons and compactons, which are manifested by particular types of singularities in the phase plane associated with the (integrated form of) the dynamical system.To proceed further, we shall need to determine the analytic continuation of the resulting solutions in the complex plane.In contrast to the KdV equation, whose sech 2 solitons have a unique extension to a single-valued meromorphic function, the solitary wave solutions of (1.1) extend to multiply-valued analytic functions, with quite complicated branching behavior.The second part of this paper is devoted to a detailed analysis of these complex analytic extensions.To determine the convergence properties, we must restrict our attention to a region supporting a single-valued extension; branching implies that the extension is not unique, but depends on how the branch cuts are arranged in the complex plane.Interestingly, the choice of branch cuts, and hence single-valued extension, a ects the convergence of the solution in the complex plane, leading to di erent non-analytic solitary wave solutions in the limit, which, nevertheless, restrict to the same peakon or compacton on the real axis.In the nal part of the paper, we shall study the properties of branch points of these solutions, and their behavior as the corresponding solutions are converging to compactons, peakons or solitary wave solutions in order to understand how singularities in uence properties of these solutions during the process of convergence.
compact support in R is denoted by C1 c = C 1 c (R).The space L p = L p (R) with 1 p 1 consists of all pth-power Lebesgue-integrable functions de ned on the real line R with the usual modi cation if p = 1.The standard norm of a function f 2 L p will be denoted by k f k p .The inner product of two functions f and g in L 2 is the integral where the overbar denotes complex conjugation.For any integer k 0 and constant p 1, the Sobolev space W k;p = W k;p (R) consists of all tempered distributions f such that f (m) 2 L p for all 0 m k.The space W k;2 is usually denoted by H k .
A (classical) travelling wave solution 1 of Equation (1.1) of wave speed c is a solution of class C 3 , having the particular form u = (x ?ct).A travelling wave solution is called a solitary wave if has a well-de ned limit lim jxj!1 (x), which is the same at both 1; the limiting value represents the undisturbed depth of the uid.A solitary wave solution, or its corresponding homoclinic orbit, is said to be analytic if the solution is a real analytic function de ned on the real axis.Analytic solitary wave solutions of integrable evolution equations, such as the KdV equation, are known as solitons, which indicates that they emerge from collisions unchanged in form, save for a phase shift; see 2], 10].
By a xed point of a dynamical system x 0 = f(t; x), where x 2 R n , we mean a point x 0 such that f(t; x 0 ) = 0 for all t 2 R. The xed point is called quasi-hyperbolic of degree one if the linearized mapping derived from the system near this point has eigenvalues with their real parts di erent from zero except one eigenvalue with zero real part.By a singularity of the dynamical system x 0 = f(t; x), we indicate a point x 0 such that f(t; x) is not analytic at (t; x 0 ) for some t in R. The xed points of system (3.2) consists of all points of the y 1 -axis.To observe properties of travelling wave solutions near each xed point (a; 0; 0) on the y 1 -axis, we let ỹ = ~ +ã = 0 @ 1 2 3 1 A + 0 @ a 0 0 1 A and substitute the transformation into (3.2),leading to the system If a < c ? 1, then there are a one-dimensional center manifold, a one-dimensional stable manifold and a one-dimensional unstable manifold near (a; 0; 0) with a unique homoclinic orbit represented by the function which is the limit of periodic cnoidal solutions of Equation (3.1).On the other hand, if a c ? 1, then there is a three-dimensional center manifold near the xed point (a; 0; 0) and (c ?1; 0; 0) is a bifurcation point of the system.
Another property of Equation (3.1) worth mentioning is that it induces a homeomorphism of (?1; c ? 1), the set of xed points having homoclinic orbits, onto the interval (c ?1; 1), the set of xed points where there are periodic orbits.This fact can be veri ed as follows.For each a < c ? 1, we substitute = + a into (3.1),integrate the resulting equation once and take the integration constant to be zero, leading to the equation ?(c ? a ? 1) + 2 2 + 00 = 0: (3.4) The dynamical system (3.4) has two xed points.One is the origin which supports a one-dimensional stable manifold and a one-dimensional unstable manifold with a unique homoclinic orbit representing the solitary wave solution expressed by (3.3).At the other xed point (2(c ? a ?1); 0), there is a two-dimensional center manifold where there exist periodic orbits converging to the homoclinic orbit at the origin as sketched in Figure 1.
Substituting = ' + 2(c ? a ? 1) into (3.4) and comparing the resulting equation ??c ? 1 ?(2c ? a ?2) ' + ' 2   2 + ' 00 = 0 with (3.1), then one may realize that periodic orbits near the point (2(c?a?1); 0) of system (3.4) can be regarded as those near the xed point (2c ? a ?2; 0; 0).In consequence, the homeomorphism : (?1; c ? 1) ?! (c ?1; 1) is naturally de ned by (a) = 2c ? 2 ?a: (3.5) Fig. 1.The phase plane of system (3.4) with a < c ? 1 This not only shows that the quasi-hyperbolic points of system (3.1) are in one-to-one correspondence with its three-dimensional center manifolds, but also indicates that for each quasi-hyperbolic point, the corresponding three-dimensional center manifold contains a sequence of periodic orbits converging to the homoclinic orbit at the quasi-hyperbolic point.We shall see that there is a similar mapping for the dynamical system obtained by reduction from Equation (1.1) which may be used to illustrate its more complicated, but more interesting properties.Now let us consider the nonlinearly dispersive Equation (1.1), and assume that 6 = 0.
Replacing u by u=( ), we reduce (1.1) to the simpler equation u t + u xxt = u x + u xxx + 3 uu x + uu xxx + 2u x u xx : (3.6) The resulting ordinary di erential equation for travelling wave solutions u(x; t) = (x?ct) of speed c is ( + c) 0 + ( + c + ) 000 + 3 0 + 2 0 00 = 0: (3.7) Using the same transformation ỹ = ~ + ã as before yields the system of equations Clearly, the set of xed points and singularities of Equation (3.7) also consists of all points of the y 1 -axis.Next, we discuss properties of each xed point or singularity (a; 0; 0) of system (3.7) in di erent cases.
Near the point ??2( 3 ( +c)+a); 0 , there exists a two-dimensional center manifold having periodic orbits converging to the homoclinic orbit at the origin as sketched in Figure 2.
Fig. 2. The phase plane of system (3.11) with ??c < a < ? 3 ( + c) in Case I When a < ??c , both (0; 0) and ( (a)?a; 0) have a two-dimensional center manifold with periodic orbits at each of the points, but they are separated by the singular point (? ?c ?a; 0) as sketched in Figure 3. On the other hand, if a 2 ??( +c ); ? 3 ( +c) , the two points (a; 0) and ( (a) ?a; 0) always stay on the right-hand side of the singular point (? ?c ?a; 0) as shown in Figure 2. The case a = ??c is the most unusual since the singular point (? ?c ? a; 0) is at the origin where a periodic orbit passes through, from which the compacton is de ned as a weak solution of Equation (3.11); while the xed point ( (a) ?a; 0) still has a two-dimensional center manifold containing periodic orbits.In this case, the interval (?( + c ); ? 3 ( + c)] on the y 1 -axis consists of xed points supporting three-dimensional center manifolds.The other two intervals, (?1; ?( + c )) and (? 3 ( +c); 1), consist of quasi-hyperbolic points, and (? 3 ( +c); 0; 0) is a bifurcation point.Unlike Case I, at the singular point (?( +c ); 0; 0), there exists only the stationary solution (x) ?( + c ) and compactons do not occur.
Fig. 6.The phase plane of Equation (3.11) in Case II with ? 1 2 ( + c ) ? 6 ( + c) < a < ? 3 ( + c) Remark.A na ve explanation for the existence of so many cuspons in this case is that the family of quasi-hyperbolic points of system (3.7)outnumbers the xed points having three-dimensional center manifolds, so that the mapping associates a great number of quasi-hyperbolic points to those of the same kind.The cuspons are present there because of the strong e ect of the singular point (? ?c ; 0; 0).Furthermore, we can use the equation ( 0 ) 2 = ? 2 ( + ( + c) + 3a) + d ( + + c + a) (3.13) derived from (3.11) by integration, where d is the integration constant, to sketch the phase plane of (3.11) for di erent values of a. Based on this, one may show that a necessary condition for a homoclinic orbit to exist at the point (a; 0; 0) is that the mapping associates the quasi-hyperbolic point (a; 0; 0) to a three-dimensional center manifold, i.e. if (a; 0; 0) is a quasi-hyperbolic point and ( (a); 0; 0) does not have a three-dimensional center manifold, then homoclinic orbits can not exist at (a; 0; 0).In contrast, there is a surplus of three-dimensional center manifolds in Case I, so that the mapping is able to associate every quasi-hyperbolic point to a three-dimensional center manifold.In addition, a homoclinic orbit is formed at each quasi-hyperbolic point because of the smaller e ect of the singular point (? ?c ; 0; 0) in this case than Case II.
Compared with system (3.7), the KdV system seems to be perfect, because the number of its quasi-hyperbolic points is balanced with the number of three-dimensional center manifolds, i.e. the mapping de ned in (3.5) is a one-to-one and onto mapping, and there are no singularities.Therefore, studying properties of one quasi-hyperbolic point of the KdV system is su cient to understand properties of other xed points in the system, whereas for system (3.7),we need to consider di erent cases in which it is also necessary to investigate xed points in di erent intervals on the y 1 -axis.
We summarize the remaining two cases to conclude this section.Case III.When > 0 and + c = ( +c)
Each a 6 = ?3 ( + c), supports a one-dimensional center manifold, a one-dimensional stable manifold and a one-dimensional unstable manifold at (a; 0; 0) without homoclinic orbits.This is not surprising since homoclinic orbits are usually accompanied by periodic orbits converging to them, but there is neither a center manifold nor a periodic orbit in this case.However we shall show that there is a cuspon at the point (a; 0; 0).On the other hand, if a = ?3 ( + c), then (a; 0; 0) is a singular point, without cuspons.
As we have seen in the above discussion, Equation (1.1) has analytic solitary wave solutions in Cases I and II, which are illustrated as homoclinic orbits in Figures 2 and  6, respectively.A question arises naturally as how these homoclinic orbits behave when the singularity (? ?c ? a; 0) is close to them.The answer is that in the rst case, the solitary wave solutions at points (a; 0; 0) converge to the compacton 0 given by (3.9) when a !??c and ??c < a < ? 3 ( +c); in the second case, the solitary wave solutions at (a; 0; 0) converge to the peakon at the xed point ? 1 2 ( ?); 0; 0 as a ! 1 then there exists an orbitally unique and analytic solitary wave solution a + a (x ?ct) for each a 2 ??? c ; ?3 ( + c) . Mreover, as a approaches ??c , the sequence of the solitary wave solutions fa + a (x)g converges to the compacton solution given in (3.9).Proof. Lt " = a+ +c , B = ?(+c+ 3a ) and B 0 = 3 ( +c )?( +c).Then Equation (3.11) for the solitary wave solution reduces to ( a + ")( 0 a ) 2 = 2 a ( B ? a ): (3.14) Using the inequality 0 a (x) B valid for all x 2 R, one may show that sequences of functions f 0 a g and f 00 a g are uniformly bounded on the real axis.Therefore, the Ascoli-Arzel a Theorem shows that, as a !??c , there exist subsequences of the families f a g and f 0 a g, without loss of generality still denoted by f a g and f 0 a g, which are uniformly convergent to a function and its derivative 0 , respectively, on any compact set of R.
Here we are relying on the fact that each a is an even function, since a is symmetric with respect to its elevation and translation invariant.Taking the limit on both sides of (3.14) as a !??c , or as " !0 leads to the equation 0 2 = 2 ( B 0 ? ) (3.15) satis ed by the function .Since lim "!0 max x2R a (x) = lim "!0 B = B 0 > 0 and each a is even, monotone on each side of the origin and exponentially decaying to zero at in nity, the limiting function is a nontrivial solution of of (3.15).Thus, satis es the equation 0 2 = ( B 0 ?).Therefore, as an even and monotone decreasing function on the positive real axis, = 0 + + c , that is to say, 0 = ??c is the compacton solution (3.9).
The corresponding result for peakons follows.then for each a 2 ?? 3 ( + c); 1 2 ( ? ) , there exists an orbitally unique and analytic solitary wave solution a + a (x ?ct).Moreover, as a ! 1 2 ( ? ), the sequence of the solitary wave solutions f a (x)g is convergent to the peakon solution (x) = ? 3 2 ( + c ) ? 2 ( + c) e ?(? ) ?1=2 jxj : (3.16) Proof.One can straightforwardly show that the rst order derivatives f 0 a g of the solitary wave solutions fa + a g at points (a; 0; 0) are uniformly bounded for all a 2 ?? 3 ( + c); 1 2 ( ? ) .Therefore, there exists a sequence of even functions monotonically decreasing on the positive axis, still denoted by f a g, satisfying the equation ( a + + c + a)( 0 a ) 2 = ? 2 a ( a + 3a + ( + c)) (3.17) and converging to the function as a ! 1 2 ( ? ).This may be derived by solving (3.17) to obtain an implicit expression of the function a and then taking the limit as a ! 1 2 ( ?); see 13] for details.
Remark.As we pointed out in the previous discussion, solitary wave solutions do not exist if + c = 3 ( + c).In case + c < 3 ( + c), we can replace u by ?u in Equation (1.1), which has the e ect of changing the sign of the coe cients and , and the wave speed c.Note that this transformation will change waves of elevation moving to the right (c > 0) into waves of depression, moving to the left.Otherwise, the conclusions in Theorems 3.1 and 3.2 also apply to the above equation.Therefore, if > 0, and a 2 ?? 3 ( + c); ?( + c ) , Equation (3.7) admits a solitary wave solution in the form a + a (x), such that the sequence of solitary wave solutions fa + a (x)g converges to a compacton as a weak solution of (3.7) when a !?( + c ).On the other hand, if < 0, then for each a 2 ? 1 2 ( ?); ? 3 ( + c) , there is a solitary wave of elevation a + a (x), such that the sequence fa+ a (x)g converges to a peakon, also as a weak solution of (3.7), as a ! 1 2 ( ? ).In either case, a satis es (3.11).In the remaining part of this paper, we shall only consider the case + c > 3 ( + c), since any result in this case can be directly applied to the case + c < 3 ( + c).
To understand how analytic solitary wave solutions converge to functions, such as compactons and peakons, having singularities on the real axis R, we shall extend solitary wave solutions mentioned in the last two theorems to functions de ned in the complex plane to study singularity distribution of these functions.This method not only provides another way to prove the last two theorems, but also makes it clear that singularities of solitary wave solutions are approaching the real axis in the process of convergence, or roughly speaking, singularities of compactons or peakons come from those of analytic solitary wave solutions, which are close to the real axis in the complex plane.This will form the subject of Part II.

3 .
Dynamical systems for solitary waves.The soliton solutions to the KdV equation can be viewed as the limits of the periodic cnoidal wave solutions; see 2], 10].Let us review this well-known fact from a dynamical systems point of view.Substituting the travelling wave solution u(x; t) = (x ?ct), for constant wave speed c, into the KdV equation (1.3), one obtains the ordinary di erential equation ?(c ? 1) 0 + 0 + 000 = 0

Fig. 3 .
Fig. 3.The phase plane of Equation (3.11) when a < ??c in Case I

Fig. 4 .
Fig. 4. The phase plane of Equation (3.11) when a < ??c in Case II

Fig. 5 .
Fig. 5.The phase plane of Equation (3.11) in Case II let the constant c be the speed of propagation of the indicated travelling wave solutions of Equation (1.1), and a the undisturbed depth.Any such solitary wave solution takes the form a + a (x ?ct), where the function = a satis es the ordinary di erential Equation (3.11) with asymptotic boundary conditions lim jxj!1 a (x) = 0. Theorem 3.1.If the coe cients of Equation (1.1) satisfy the inequalities 6