## Abstract

Multi-pulse homoclinic orbits of Hamiltonian systems on can be classified by a sequence of integers. In this paper, we find the surprising result that this string of integers encodes the value of the Maslov index of the homoclinic orbit. Our results include a computable formulation of the Maslov index for homoclinic orbits and a robust numerical method for the evaluation of the Maslov index.

## 1. Introduction

Homoclinic orbits are important in Hamiltonian dynamical systems. They can be organizing centres for chaos, and in the case where they are steady-state solutions of an evolutionary partial differential equation (PDE) they represent localized solutions such as solitary waves. Of interest in this paper are multi-pulse homoclinic orbits.

A multi-pulse homoclinic orbit is firstly a homoclinic orbit. The ‘multi-pulse’ nature indicates multiple maxima and minima in the graph of the function. An important open question is how to distinguish between two multi-pulse homoclinic orbits. A universal classification of multi-pulse homoclinic orbits has yet to emerge. However, for a class of autonomous Hamiltonian systems on , Buffoni *et al.* (1996) have introduced a precise classification based on a sequence of integers. Multi-pulse homoclinic orbits are labelled by **n**(ℓ_{1},…,ℓ_{n−1}) where **n** is the modality (the number of major local extrema) and ℓ_{1},…,ℓ_{n−1} are related to the number of minor bumps between consecutive extrema. A precise definition is given in Buffoni *et al.* (1996). Hereafter, this classification is called the BCT classification.

An important topological invariant of any orbit of a Hamiltonian system is the *Maslov index*. A precise definition of the Maslov index in this context is given in §2. We have found that the Maslov index of a homoclinic orbit is encoded in the BCT classification as
1.1
where *n*_{even} (*n*_{odd}) is the number of even (odd) integers in the sequence ℓ_{1},…,ℓ_{n−1}. This observation is numerical. The results are obtained by explicit computation using a new numerical algorithm for computing the Maslov index.

The class of Hamiltonian systems of interest is steady solutions of the PDE
1.2
where *P* is a real parameter. Steady solutions are orbits of the one-dimensional equation (ODE)
1.3
This ODE has been extensively studied because of its importance in pattern formation. It is called the canonical equation in the book by Peletier & Troy (2001), and it is the ODE that forms the basis of the theory of Buffoni *et al.* (1996). The ODE (1.3) can be characterized as a Hamiltonian system and the formulation used here is recorded in appendix A.

The PDE (1.2) arises in many applications: beam buckling, pattern formation – where it is called the one-dimensional Swift–Hohenberg equation (Burke & Knobloch 2007), thin-film flows and a variant (an additional space derivative is added to the right-hand side) arises in the theory of capillary-gravity water waves, called the fifth-order KdV equation. A review of equations of this type is given by Champneys (1999).

The linearization of equation (1.2) about a multi-pulse solution satisfying equation (1.3) and taking exponential in time solutions *ϕ*(*x*,*t*)↦e^{−λt} *ϕ*(*x*) leads to the linear ODE
1.4
This ODE can be reformulated as a standard linear Hamiltonian system on as
1.5
where **B**(*x*,*λ*) is a symmetric matrix depending smoothly on *x* and *λ* with the property
1.6
and
1.7
The matrix **B**(*x*,*λ*) is defined in equation (A 2).

The parameter *λ* serves two purposes. It is a stability exponent for the PDE, but it is also a device for assuring genericity (see §2*a* for discussion). We will be primarily interested in the limit *λ*→0.

The Maslov index is most familiar in the literature in the context of closed orbits, e.g. the case of equation (1.5) with periodic coefficients, because of its importance in semi-classical quantization (cf. Arnold 1967; Littlejohn 1986; Littlejohn & Robbins 1987; Robbins 1991; Pletyukhov & Brack 2003, and references therein). The Maslov index of homoclinic orbits was first introduced by Jones (1988) and Bose & Jones (1995) for one-pulse homoclinic orbits. We turn this definition into a computable formula by introducing an explicit intersection form, following Robbins (1991, 1992) and Robbin & Salamon (1993). Chen & Hu (2007) have recently introduced a formulation of the Maslov index for homoclinic orbits. They give two constructions. The first is based on an intersection index and therefore is a generalization of the Bose–Jones definition. Their second definition is based on a relative Morse index of equation (1.5) considered as an operator on the real line. It is the intersection index formulation that is most useful for numerics.

The Maslov index for a homoclinic orbit can also be defined by approximating the homoclinic orbit by a periodic orbit and then taking the limit as the period of the orbit goes to infinity. The existence of this Maslov index was recently proved in Chardard (2007). Both definitions are used in the numerical computation to double check the results.

The paper has three parts: derivation of a computable formula for the Maslov index, a numerical algorithm on exterior algebra spaces for computing the Maslov index and results for multi-pulse orbits of equation (1.3).

## 2. The Maslov index of a homoclinic orbit

A subspace span{*ξ*_{1},*ξ*_{2}} of is a Lagrangian subspace if *ξ*_{1} and *ξ*_{2} are linearly independent and 〈**J***ξ*_{1},*ξ*_{2}〉=0. Here and throughout 〈·,·〉 is the standard inner product on . The stable and unstable subspaces of equation (1.5) are Lagrangian subspaces.

The matrix and is defined in equation (A 3). The characteristic polynomial of is
For *P* < +2 and *λ*=0, all four roots have non-zero real parts and this property will persist for small *λ* (and we are only interested in small *λ* in this paper). In fact, two eigenvalues have positive real part and two have negative real part. Associated with the eigenvalues with positive (negative) real part is a two-dimensional unstable (stable) subspace. Denote the stable subspace of by
where *ξ*_{1} and *ξ*_{2} are the eigenvectors associated with the eigenvalues of with negative real part
2.1
with appropriate modification if μ_{1}=μ_{2}. *E*^{u}(*λ*) is defined analogously. It is easy to verify that E^{u}(*λ*) and E^{s}(*λ*) are Lagrangian subspaces, and their *x*-dependent extensions are also Lagrangian.

Let **U**^{+}(*x*,*λ*) be a 4×2 matrix whose columns span the *x*-dependent unstable subspace and so **U**^{+}(*x*,*λ*)→0 as . **U**^{+}(*x*,*λ*) is a path of Lagrangian planes. The Maslov index is defined as the signed count of the intersections of the image of **U**^{+} with *E*^{s} as *x* goes from to . This definition of the Maslov index for homoclinic orbits is equivalent to the definition introduced in Jones (1988) and Bose & Jones (1995), although a computable expression for the intersection form is required.

A point *x*_{0} is called a point of one-dimensional intersection between Image(**U**^{+}) and *E*^{s} if
for some non-zero vector . Clearly this implies that
at the intersection for some and . A one-dimensional intersection is regular at *x*_{0} if the crossing is transversal as *x* is varied. To test for transversality, an intersection form is used.

Formulas for the intersection form have been given in Robbins (1991, 1992) and Robbin & Salamon (1993). These representations are equivalent (modulo a choice of orientation) and they are based on the following formula. At a point of one-dimensional intersection *x*_{0}, the crossing form is defined as
where vol is the chosen volume form, which is fixed throughout to be
with .

When *Γ*(**U**^{+},E^{s},*x*_{0})≠0, the intersection is said to be *regular*. The *sign of the intersection* is defined as the sign of *Γ*(**U**^{+},E^{s},*x*_{0}).

Using the differential equation and the representation of *ξ* in equation (2.1), this is
2.2

We are now in a position to define the Maslov index of the path **U**^{+}. Suppose that
and for assume that the intersections between Image(**U**^{+}) and *E*^{s}(*λ*) are one-dimensional (higher-order intersections can also be accounted for but will not be needed here) and regular. Then the Maslov index of the path **U**^{+} is
2.3
where the sum is over all points of intersection .

The *Maslov index of the homoclinic orbit* is defined as the limit as *λ*→0^{+},
2.4

### (a) The role of *λ*

The appearance of *λ* in the formulation appears odd, as we are interested in the Maslov index of homoclinic orbits and this index should be defined purely in terms of the steady equation (1.3). However, the parameter *λ* is of interest for two reasons. First, it is a stability exponent for the time-dependent equation (1.2). While the issue of stability is only briefly remarked on in this paper, the stability of solitary waves is one of the motivating factors in the study of the Maslov index (cf. Jones 1988; Bose & Jones 1995; Chardard *et al.* in press).

The second reason that *λ* is useful as a numerical device. When *λ*=0, there is a bounded solution of equation (1.5) ( in the notation of appendix A), and so the Maslov index jumps by 1 at this point. By perturbing *λ* slightly, this property is eliminated. Hence the numerical computations are carried out with small *λ*. Taking the limit *λ*→0^{−} differs from *λ*→0^{+}, but it is merely a convention to decide which limit to take. Here we have opted for the + convention (2.4).

In principle the Maslov index (2.3) is straightforward to compute. We fix *λ* near 0^{+} and integrate the unstable subspace of (1.5) from *x*=−*L* to *x*=+*L* for some large *L* and sum the signed intersections. However integration of the unstable subspace is numerically unstable! To avoid this difficulty, the induced equation on is integrated and this approach is numerically stable (Allen & Bridges 2002; Chardard *et al.*2006, in press).

## 3. Exterior algebra representation of Maslov index

The advantage of an exterior algebra representation is that two-dimensional subspaces of become lines in the exterior algebra space. This strategy reduces the numerical integration to a problem similar to integration on , and then numerical integration is trivial. The details of numerical integration on exterior algebra spaces in this context are given in Chardard *et al.* (in press) and only the basic details are given here.

is a six-dimensional vector space spanned by all non-trivial two-vectors of the form **e**_{i}∧**e**_{j}, where *i*,*j*=1,…,4. The orthonormal basis induced from the standard basis of is
3.1
Any can be represented in the form . An element of does not necessarily represent a two-plane. A point represents a two-plane if
This submanifold of the projectification of is the Grassmannian of two-planes in , . A two-plane is Lagrangian if, in addition, it satisfies
where **ω**=**e**_{1}∧**e**_{3}+**e**_{2}∧**e**_{4} is the two-form associated with **J**.

The practical implementation involves constructing an induced ODE on . Given the linear system **u**_{x}=**A**(*x*,*λ*)**u** on with **A**=**J**^{−1}**B**(*x*,*λ*), there is an induced linear system on ,
Here **A**^{(2)}(*x*,*λ*) is a 6×6 matrix whose entries are linear functions of the entries of **A**(*x*,*λ*). A formula for the entries is given in §2 of Allen & Bridges (2002) and the induced matrix associated with equation (1.5) is given in appendix A.

In this setting, the stable subspace at infinity is represented by the two-form *ξ*_{1}∧*ξ*_{2} and the path of unstable subspaces is represented by a two-form which will be denoted by **U**^{+}(*x*,*λ*). It satisfies
3.2
where represents the unstable subspace E^{u}(*λ*) and σ_{+}(*λ*) is the sum of the two eigenvalues of with positive real part.

An intersection between **U**^{+} and *E*^{s}(*λ*) can be described as follows. **U**^{+} has a non-trivial intersection with E^{s} if and only if there exists with *α*≠0 and
3.3
If **U**^{+}(*x*,*λ*)∧(*α*_{1}*ξ*_{1}+*α*_{2}*ξ*_{2})=0 implies *α*=0, then we say that **U**^{+}(*x*,*λ*) and E^{s}(*λ*) are transverse. At a one-dimensional *regular* intersection *α*≠0, but it is a one-dimensional subspace of . Intersections can also be checked by monitoring the sign changes of the four-form **U**^{+}∧*ξ*_{1}∧*ξ*_{2}, but the test based on equation (3.3) is needed for construction of the crossing form.

In the exterior algebra setting, a representation of the crossing form is
3.4
To verify that this is equivalent to equation (2.2), note that
Hence **ω**∧*ξ*∧**A***ξ*=**ω**∧*ξ*∧**J**^{−1}**B***ξ*=〈*ξ*,**B***ξ*〉 vol. The formula (3.4) has an interesting geometric interpretation. At a regular intersection, the two-plane *ξ*∧**A***ξ* is *not* a Lagrangian plane. It is in the complement to the Lagrangian in .

Suppose that is transverse to E^{s}(*λ*) and suppose all the intersections are one-dimensional and regular, then
The Maslov index of the multi-pulse homoclinic orbit is then obtained by taking the limit *λ*→0^{+}.

## 4. An algorithm for computing the Maslov index

An algorithm for computing the Maslov index based on the exterior algebra representation is constructed as follows. Fix *λ*. Compute a basis for E^{s}(*λ*) and E^{u}(*λ*). Earlier, E^{s}(*λ*) was expressed as span{*ξ*_{1},*ξ*_{2}}. Here the stable and unstable subspaces will be represented in the exterior algebra representation. This just means solving
with σ_{+}(*λ*) (σ_{−}(*λ*)) the *sum* of the two eigenvalues of with positive (negative) real part. Then ζ^{−}(*λ*) (ζ^{+}(*λ*)) represents E^{s}(*λ*) (E^{u}(*λ*)).

Integrate equation (3.2) with the exponential growth e^{σ+(λ)x} factored out1
4.1
with
4.2
for some large value of *L* (typically we have used *L*=25). For the numerical integration, a standard fourth-order explicit Runge–Kutta algorithm is used. All the codes are written in Matlab. Further details of the numerics including listings of the Matlab codes can be found in the thesis of Chardard (2009).

Intersections are detected and counted as follows. Let
*Y*_{1} and *Y*_{2} are three-forms and is a four-dimensional vector space, so it is isomorphic to (this isomorphism can be explicitly constructed but is not needed). An intersection occurs when *Y*_{1} and *Y*_{2} are linearly dependent and, viewed as vectors in , *Y*_{1} and *Y*_{2} are linearly dependent if and only if *Y*_{1}∧*Y*_{2}=0. Define
4.3
Then *Y*_{1}∧*Y*_{2}=0 is equivalent to *y*(*x*,*λ*)=0, or numerically when *y*(*x*,*λ*)<ε for some small ε>0.

At each intersection, *α*_{1} and *α*_{2} are computed by solving *α*_{1}*Y*_{1}+*α*_{2}*Y*_{2}=0. Then *α* is used to construct *ξ* and the crossing form is evaluated. To summarize: integrate equation (4.1) with initial condition (4.2) from *x*=−*L* to *L*. Monitor *y*(*x*,*λ*) in equation (4.3) and when *y*(*x*,*λ*)≈0, compute the sign of the crossing form and add the appropriate sign to the Maslov counter. The Maslov index is then the value of the Maslov counter at *x*=*L*.

## 5. Computing the Maslov index for orbits of equation (1.2)

Apply this theory to the system (1.5) associated with equation (1.2) linearized about a multi-pulse homoclinic orbit. The induced matrix on is given in equation (A 4). The system at infinity is hyperbolic for all real *λ* satisfying *λ*<1 (when *P*<0) and (when *P*>0), and has two eigenvalues with positive real part and two with negative real part. The eigenfunctions associated with E^{s}(*λ*) and E^{u}(*λ*) are easily calculated.

Before computing branches of multi-pulse homoclinic orbits in the BCT classification, we sketch the properties of the classification. A homoclinic orbit is classified according to the number of times its path in configuration space encircles the origin. The orbit is labelled by **n**(ℓ_{1},…,ℓ_{n−1}), where *n* is a natural number called the modality, the number of large local maxima, and ℓ_{k} is twice the number of encirclings of the origin of configuration space, between consecutive local maxima. A family *ψ*_{P} of solutions of equation (1.3) is said to have a mode at *s*_{i,P} if
A multi-pulse homoclinic orbit is said to have type **n**(ℓ_{1},ℓ_{2},…,ℓ_{n}) if it has *n* modes at the points *s*_{1,P},*s*_{2,P},…,*s*_{n,P}, and
and the number of zeros of in [*s*_{i,P},*s*_{i+1,P}] is equal to 2ℓ_{i}. BCT have conjectured that there is a unique family of each type, up to a space translation.

An important property of this classification is that it is precise only in the limit *P*→−2. Therefore, the strategy for computing a multi-pulse orbit of equation (1.3) in the BCT classification is to start with an orbit near *P*=−2 and then continue it in *P*. Examples of bimodal multi-pulse orbits are shown in figure 1. The **2**(ℓ) pulses in this figure are all symmetric. They were computed using the shooting algorithm of Champneys & Spence (1993). Indeed, figure 1 was inspired by fig. 4*a* in Champneys & Spence (1993). An example of a symmetric multi-pulse orbit of modality **5** is shown in figure 2.

Asymmetric solutions are computed using a Fourier method (approx. the homoclinic orbit by a periodic solution of large wavelength). An example of an asymmetric multi-pulse orbit of type **3**(3,1) is shown in figure 3. The Matlab codes used are listed in the appendix of Chardard (2009).

We now proceed to compute the Maslov index for a series of multi-pulse orbits. Fixing a multi-pulse orbit in the BCT classification, the Maslov index is computed using the algorithm in §4. To double check the computations, we also computed the Maslov index by approximating the multi-pulse orbit by a periodic solution of long wavelength, and then using the formula in Chardard *et al.* (2006) and Chardard (2007). See Chardard *et al.* (in press) for other algorithms for computing the Maslov index. The results for a range of orbits in the BCT classification are listed in table 1.

The table shows that all computed orbits satisfy the formula (1.1), and so it is reasonable to conjecture that it is true for all multi-pulse orbits in the BCT classification.

There are other examples in the literature of multi-pulse homoclinic orbits that can be identified with the BCT classification and so their Maslov indices can be predicted. The homoclinic orbits found by Champneys & Toland (1993) are of type **2**(ℓ) and so they have Maslov index 3 (if ℓ is even) or 4 (if ℓ is odd). They were proved to exist for *P*∈(−2,−2+*ϵ*). Numerical results of Buffoni *et al.* (1996) suggest that homoclinic orbits of the type **n**(ℓ_{1},…,ℓ_{n−1}) exists for all *n*≥2 and for all *P*∈(−2,1.5]. The homoclinic orbits found by Buffoni (1995) are of type **n**(1,…,1) and hence they have Maslov index 2*n*, and they were shown to exist for all *P*∈(−2,0]. One can deduce from this that homoclinic orbits of the ODE (1.3) exist with Maslov index of every natural number greater than or equal to two.

## 6. Bifurcation and coalescence

The branches of multi-pulse homoclinic orbits that start at *P*=−2 appear to end at some value of *P* < 2. Buffoni *et al.* (1996) identified two types of termination point: *coalescence* and *bifurcation*. A coalescence corresponds to a turning point and can occur along a symmetric or asymmetric branch. A ‘bifurcation’ in this context is a point where a symmetric branch changes type and has a pitchfork bifurcation to a asymmetric branch. A theory for bifurcation and coalescence using a Lyapunov–Schmidt reduction has been developed by Knobloch (1997). In this section, we look at the implications of coalescence and bifurcation for the Maslov index. We will concentrate on one example that is illustrative.

The numerics also showed that both bifurcation and coalescence often occur along the same branch. A schematic of fig. 17*c* in Buffoni *et al.* (1996) is shown in figure 4. This scenario is an ideal setting to test the implication of the Maslov index. Indeed, we have found a surprising result. Consider the case in figure 4. We can apply the formula (1.1) to deduce that the branch **4**(2,1,2) has Maslov index 6. We confirmed this value numerically. Then after the coalescence point, the computations show that the Maslov index jumps to 5. Now, start from the branch **2**(1). The formula and numerics show that the Maslov index is 4. Similar use of the formula and computation show that the Maslov indices of the asymmetric **3**(1,2) and **3**(2,1) branches are also 5. The numerically computed multi-pulse homoclinic orbits emanating from the bifurcation point are shown in figure 5.

These observations show an interesting anomaly in the BCT classification. The short branch between the coalescence point and the bifurcation point is not classifiable by the BCT scheme. On the other hand, this observation is consistent with the theory of BCT. A branch of multi-pulse homoclinic orbits can only be BCT classified if it can be continued to *P*=−2. It is precisely the branch between the coalescence and bifurcation points that cannot be continued to *P*=−2. All other branches in figure 4 can be continued to *P*=−2. These observations have been obtained numerically. By adapting the theory of Knobloch (1997), Chardard (2009) has sketched an argument showing that indeed the Maslov index jumps by 1 at each bifurcation point and at each coalescence point.

The property of coalescence and bifurcation occurring close together along a branch appears to be pervasive, and so there will be many gaps where the multi-pulse homoclinic orbits are not classifiable by the BCT scheme (see fig. 24 of Buffoni *et al.* 1996). However, all these orbits have a well-defined Maslov index.

## 7. Remarks on the Morse index and stability of solitary waves

Let Then equation (1.4) can be written as . Informally, the Morse index of is the number of negative eigenvalues of . To be precise, the function space needs to be identified and the spectrum decomposed. Here, just a rough idea of the connection between the Morse and Maslov indices is given. In general, the Morse and Maslov indices are not equal. Indeed, one can construct examples where the Morse index is infinite, but the Maslov index is finite. On the other hand, the operator has a nice structure. It has a monotonicity property that assures that the Maslov and Morse indices are equal (this is proved in Chardard 2009).

With that observation, the role of the Maslov index in stability is quite remarkable. For example, in the Swift–Hohenberg equation (1.2), viewing the multi-pulse homoclinic orbits as stationary localized solitary waves, the Maslov index equals the number of unstable eigenvalues. Therefore, one can conclude that the higher modality multi-pulse solitary waves—indeed all solitary waves in the BCT classification—are linearly unstable solutions of (1.2), and the higher the modality, the more unstable it is.

When this theory is applied to the fifth-order KdV equation, however, the results are more interesting because high Maslov index solitary waves can still be stable (see Chardard *et al.* in press for results in this direction).

## Appendix A. Hamiltonian formulation

The ODE (1.3) can be formulated as a Hamiltonian system in many ways. The Hamiltonian formulation used in the numerics is recorded here. Let
and define
Then equation (1.3) is represented by the Hamiltonian system
A 1
where **J** is the standard symplectic operator defined in equation (1.7).

Let be a solution of equation (1.3) and its associated solution of equation (A 1), then
Then the linearization of equation (A 1) about is
where is the Hessian of *H* evaluated at . Define
A 2
The linear Hamiltonian system **J****u**_{x}=**B**(*x*,*λ*)**u** with is the main object of the study in this paper. The ‘system at infinity’ is defined by
A 3
Define **A**(*x*,*λ*)=**J**^{−1}**B**(*x*,*λ*). Then the matrix induced from **A**(*x*,*λ*) on is
A 4
For any , this matrix is defined by
Details of the construction of induced matrices are given in Allen & Bridges (2002).

## Footnotes

↵1 Factoring out the exponential growth is purely a numerical device to ensure stable integration.

- Received March 26, 2009.
- Accepted June 5, 2009.

- © 2009 The Royal Society