NLS Bifurcations on the bowtie combinatorial graph and the dumbbell metric graph

We consider the bifurcations of standing wave solutions to the nonlinear Schr\"odinger equation (NLS) posed on a quantum graph consisting of two loops connected by a single edge, the so-called dumbbell, recently studied by Marzuola and Pelinovsky. The authors of that study found the ground state undergoes two bifurcations, first a symmetry-breaking, and the second which they call a symmetry-preserving bifurcation. We clarify the type of the symmetry-preserving bifurcation, showing it to be transcritical. We then reduce the question, and show that the phenomena described in that paper can be reproduced in a simple discrete self-trapping equation on a combinatorial graph of bowtie shape. This allows for complete analysis both by geometric methods and by parameterizing the full solution space. We then expand the question, and describe the bifurcations of all the standing waves of this system, which can be classified into three families, and of which there exists a countably infinite set.


1.
Introduction. Linear and nonlinear waves on quantum graphs have recently gained a lot of attention in mathematical literature as a nontrivial generalization of well-known systems and because they possess novel mathematical properties not possible in simpler topology [7,8]. In physics they may arise in optical systems, Bose-Einstein condensates, and in the study of large molecules such as carbon nanotubes [10,13,26,31]. In the nonlinear regime they lead to bifurcations and, in the case that one or more edges extends to the point at infinity, to interesting questions about the existence of minimizers [2,3,4,5,6,11,12,32,35].
While there is an infinite variety of graphs, a lot of recent work has gone into a few simple ones. The tadpole graph consists of a half-line joined to a loop at a single vertex [33]. The lollipop graph is similar, with the half-line replaced by a finite line segment [8]. Star graphs consist of a finite number of half lines or line segments that meet at a single vertex [1].
In a recent paper, Marzuola and Pelinovsky study the bifurcations of stationary solutions to the cubic nonlinear Schrödinger equation posed on a dumbbell-shaped quantum graph [27], in which two loops are connected via a line segment. At small L 2 norm, the ground state, i.e. the lowest energy solution with that norm, is constant in the spatial coordinate. They show that this state undergoes two bifurcations as the solution amplitude is increased. The first is a symmetry-breaking or pitchfork bifurcation and the second they call a "symmetry preserving" bifurcation, producing a solution localized on the central edge of the graph. At the symmetry-breaking bifurcation, the asymmetric solution becomes the ground state.
We show that all of these phenomena are present in a simpler system, a discrete NLS equation on a graph with five vertices. We show that in this simpler system the "symmetry preserving" bifurcation is actually a transcritical bifurcation, and, thus, contains a branch of solutions not present in the bifurcation diagram shown in [27]. This is interesting, because Yang, in Ref. [39], only reported transcritical bifurcations by constructing an asymmetric potential, whereas the dumbbell graph is symmetric; see Remark 3.1 for more details. We then show, via a perturbation calculation, and via numerical computation that the same is true for the problem on the quantum graph, which demonstrates the existence of the branch not found in [27]. Further, we perform an extensive numerical study, enumerating all the branches of solutions that exist in a certain part of parameter/solution space.
1.1. Organization. The paper is organized as follows. The rest of this section contains a brief introduction to NLS on combinatorial and quantum graphs and a summary of recent and related results, as well as a brief review of bifurcation theory. The section closes with a summary of the Marzuola and Pelinovsky's bifurcation diagram, explaining why it must be missing something. Section 2 introduces and completely analyzes a simple combinatorial graph model that reproduces the two bifurcations seen in [27] for the dumbbell quantum graph. Section 3 returns to the dumbbell quantum graph. It begins by summarizing some additional background information, including Yang's framework for classifying bifurcations in NLS-like systems. It then describes arguments using perturbation theory and numerical continuation that show that the symmetry preserving bifurcation is indeed transcritical and to recover the half-branch of solutions not discussed in [27]. Section 4 describes a singular limit of the dumbbell graph in which the length of edges e 1 and e 3 approaches zero. This demonstrates an interesting feature of the loop-shaped edges. Section 5 contains a full classification and computational enumeration of all types of standing waves possible on the dumbbell graph. This is based on the observation that there are two types of behavior possible at the two vertices, and therefore three types of standing waves, as the two vertices can both have the first behavior, both have the second, or be different. After making this classification, we use a combination of exact solution formulae and numerical continuation to draw bifurcation diagrams for the three types of solutions. In fact, each of the three families contains a countably infinite set of branches. Section 6 wraps up and discusses some possible future directions.
1.2. NLS on combinatorial and metric graphs. The focusing cubic nonlinear Schrödinger equation (NLS) i∂ t Ψ = ∆Ψ + 2 |Ψ| 2 Ψ, (1.1) subject to appropriate boundary conditions or conditions at infinity, is perhaps the most-studied system in nonlinear waves, arising as an envelope equation in systems where dispersion is roughly balanced by nonlinearity. It is the simplest PDE with both these features, and thus is widely used as a model problem. Many variants of this equation have been studied in order to explore a wide range of phenomena.
One of the simplest variants is the discrete nonlinear Schrödinger equation (DNLS) in which the spatial derivative is replaced by a discrete second difference operator iu n = u n−1 − 2u n + u n+1 + |u n | 2 u n . (1.2) If the total number of sites is small this system is also known as the discrete self trapping (DST) system, which has a long history [17,18,24]. A DST system describing the evolution of a vector u may be written, more generally, as where L is a symmetric matrix and N (u) is a nonlinearity we take to be local and cubic, N (u) n = |u n | 2 u n . This system may be interpreted as a Hamiltonian, with position vector q = u ∈ C N and momentum vector p = iū (such overbars will denote complex conjugates in the remainder). It is then easy to check that the Hamiltonian for this system is simply (1.4) The DST system (1.3) is then equivalent to Hamilton's equations: NLS may be generalized to graphs in a number of ways. Before doing so, we first set notation, briefly summarizing that of Berkolaiko [8]. Let Γ = (V, E) be a combinatorial graph, i.e. a finite collection of vertices V = {v n , n = 1, . . . , N } connected pairwise by a collection of edges If Γ has no self-directed edges, i.e. no edges of the form (v i , v i ), the incidence matrix is defined to be the N × M matrix E in which if edge m points to vertex n, −1 if edge m points from vertex n, 0 otherwise.
Let there be a positive weight w m associated to edge e m and let W be a diagonal matrix with Ω mm = w m > 0. If unstated, then W is assumed to be the identity matrix. Then the (weighted) Laplacian is a negative definite matrix, defining a DST system (1.3) with L = −∆ Γ . While a directed graph is necessary for forming E and thus ∆ Γ , the matrix ∆ Γ is unchanged if the direction of any edge is interchanged. A metric graph is a directed graph in which a length m has been assigned to each edge e m , as well as a coordinate x m with 0 < x m < m which increases in the specified direction of the edge. A function f (x) defined on the graph is simply a collection of functions f m (x m ) defined on the individual edges e m such that f (x)| em = f m (x m ). A Schrödinger operator L defined on such functions, although we will set the potential along each edge to V m (x m ) = 0. Defining the function space on which L acts requires a boundary condition at the vertices. The graph, operator, and boundary conditions together define a quantum graph, and we may extend the definition of equation (1.1) to quantum graphs. The most common boundary condition is the Kirchhoff boundary condition, which states that the solution is continuous at the vertices, i.e. letting Ψ| em = ψ m , where V n is the set of all edges incident on vertex v n , and further, that the total flux through any vertex is zero, i.e where σ m = 0 if v n is the initial point on edge e m and σ m = 1 if it is the final point. Both ends of an edge that connects a single vertex to itself contribute to this sum. Observe that for leaf vertices, connected to only a single edge, the Kirchoff boundary condition reduces to a Neumann condition. The set V may contain a point at infinity, and special edges called leads that connect the finite points with the point at infinity. Such an edge must have = ∞.
The standard norms and function spaces are defined straightforwardly, e.g. the L p norms and the related Sobolev norms defined similarly. In particular, we define the power to be (1.7) The L 2 inner products is defined by Cubic NLS on the quantum graph is thus which is defined on the energy space E(L) consisting of all H 1 functions on Γ satisfying the boundary conditions (1.6b) and (1.6c). This evolution conserves the energy . This paper considers standing wave solutions to the NLS equation of the form Ψ(x, t) = Φ(x)e iΛt . Assuming Φ to be real valued, these solve For all Λ, Φ = 0 is an exact solution. For Λ < 0, there are additional solutions at which we refer to as the constant solutions.
1.3. Bifurcation theory. As a principal aim of this paper is to distinguish among different bifurcations that occur in the various families of standing waves, we include a brief review of the relevant bifurcation theory. The most basic question in bifurcation theory concerns fixed points of the differential equation which is interpreted as as a one-parameter family of ODE for the evolution of a(t), parameterized by Λ. If a 0 is a fixed point of equation ( The solution that exists on both sides of the bifurcation is invariant to the symmetry, while the two newly-created solutions are not invariant to the symmetry, together they form a group orbit of that symmetry. The canonical example is f (a, Λ) = Λa−a 3 . At the bifurcation point, the stability of the branch of fixed points changes. While in the basic examples above, the unknown a is scalar, the same bifurcations are found in multidimensional and even infinite-dimensional systems. Yang has computed conditions under which the different types of bifurcations occur in the stationary NLS equation [39]. His results, which go through unaltered in the quantum graph case, are described in detail in Section 3.2. A second theorem by Yang describes the arrangement of the bifurcation curves for NLS when the frequency Λ is plotted against the power Q, defined in equation (1.7); see Remark 3.2 below for a more complete discussion. The second row of Figure 1 Ref. [27] considers stationary solutions to equation (1.9) on a dumbbell-shaped graph with two vertices and three edges, as displayed in Fig. 1.2. Two of this graph's edges are loops that connect the two vertices to themselves, and the third connects them. The authors make the symmetry assumption 1 = 3 which, without loss of generality they take to be 1 = 3 = 2π, and let 2 = 2L, parameterized by −π < x 1,3 < π and −L < x 2 < L.  We note that symmetries of the graph play a role in the solutions to a quantum graph NLS system. The linear and nonlinear Schrödinger equation on the underlying dumbbell metric graph have the following three reflection symmetries, which we state somewhat informally:   Ref. [27] considers the stability of the constant solution (1.10), using its frequency Λ as a bifurcation parameter. It shows that the constant solution Φ 0 undergoes a pair of bifurcations with decreasing Λ. The first is a symmetry-breaking pitchfork bifurcation, and occurs at Λ = −Ω 2 1 /2 where Ω 2 1 is the smallest eigenvalue with an even eigenfunction (see Section 3.1). While the solution Φ const is invariant under R 2 , the two newly-created states are not; instead forming a group orbit for that symmetry. More concretely, two nonlinear eigenfunctions Φ left and Φ right emerge, each asymmetric, but with Φ left = R 2 (Φ right ). The second bifurcation, which the authors call a "symmetry-preserving" bifurcation, takes place at Λ = −ω 2 1 /2, where ω 2 1 is the eigenvalue corresponding to the first non-constant even-symmetric eigenfunction. Here they find the emergence of a symmetric solution that is concentrated along the central edge e 2 of the dumbbell as Λ → −∞.
The above statement is correct but incomplete. "Symmetry-preserving" bifurcation is not one of the generic categories, so the question remains to identify it. A bifurcation diagram computed in Ref. [27] for L = π (in the notation of [27] it is L = π/2) is shown in Fig. 1.3, showing the two bifurcations discussed in that paper. Observe that this diagram is incomplete: the branch labeled "Centered" does not meet the branch labeled "Constant." Arguments given both in Ref. [27] and summarized later in this paper allow us to calculate in closed form the bifurcation value where the constant state bifurcates. The symmetry-breaking bifurcation occurs at −Ω 2 1 /2 ≈ −0.036 where 3 ), which agrees well with the calculated bifurcation diagram. The symmetry preserving bifurcation takes place at −ω 2 . This location is shown in the figure to be far from the spot where the two curves appear to meet. The present paper was initially motivated by two questions: to understand what exactly occurs at the symmetry preserving bifurcation and to determine whether this effect is novel to the dumbbell quantum graph or if, instead, it occurs in simpler systems. The answers are simple: the symmetry-preserving bifurcation is of transcritical type, and the same phenomenon can be found in a significantly simpler system, the bowtie combinatorial graph. It appears that the closed loops in both the dumbbell and the bowtie contribute to the transcritical nature of the second bifurcation. In a simpler related problem, the Schrödinger equation on an interval with Neumann boundary conditions, all the bifurcations are pitchforks. This distinction is due to a subtle breaking of symmetry by the loops, as we shall discuss. Moreover, on the lollipop graph, featuring one loop and one straight edge, all of these bifurcations are transcritical.
Remark 1.1. The central finding of Ref. [27] concerned the ground state and was subsequently found to be mistaken. The ground state of the quantum graph is defined to be the lowest energy state with a given L 2 norm, i.e. the solution of The ground state for the DST system is defined analogously for the DST with the Hamiltonian (1.4) taking the role of the energy, in particular, it is the solution to This mistake has since been corrected by its authors online [28]. Upon its creation in the symmetry-breaking bifurcation, the asymmetric loop-centered state is the ground state of the system. The analysis of [27] seemed to show, incorrectly, that for sufficiently large values of L 2 , the centered solution concentrated on the central edge of the graph has the lower energy. While this solution is a local constrained minimizer of the energy in equation (1.13), while the loop-centered solution is the global minimizer.  Setting the weights to one, the negative of the Laplacian matrix is with eigenvalues λ 1 = 0, λ 2 = 5, λ 3 = 1, and λ 4 = λ 5 = 3 (numbered in this strange order purposely) and associated normalized eigenvectors,

NLS BIFURCATIONS ON THE DUMBBELL METRIC GRAPH 2211
The DST system on this graph is invariant to left multiplication by the three permutations, in cyclic notation: 1 These symmetries are in exact analog with (1.12) and are labeled analogously. The vectors in the space Like the the dumbbell quantum graph, the bowtie DST has a constant solution Linearizing about this solution, we find that it undergoes bifurcations at Ω = − 1 2 and Ω = − 5 2 . To understand these bifurcations and the more detailed bifurcation structure, we turn to parameterization.

2.1.
Parameterization of solution branches on S 2 . In this section, we follow [18] and use simple algebraic methods to enumerate all the relative fixed points of the discrete bowtie system that lie on the invariant subspace S 2 , i.e. we look for stationary solutions of the equation (1.3) with Laplacian matrix given by (2.1) of the form The reduced system consists of the three equations We will derive several branches of solutions. For later convenience, we will assign each branch a number. We first derive two simple solutions: • Branch 1 is the constant state a = b = c, the continuation of the eigenvector v 1 , for which Ω = −a 2 . The squared 2 norm for this solution is just Q = 5a 2 , so that Ω = −Q/5. • Branch 2 satisfies c = −a, b = 0, the continuation of eigenvector v 3 , with Ω = 1 − a 2 . This solution has Q = 4a 2 , so Ω = 1 − Q/4. We parameterize all the remaining solutions as follows. Subtracting equation Letting either factor equal zero yields a solution. Setting the first factor to zero yields a = c, and the solution lies on the invariant subspace S 1 . This reduces system (2.2) to Subtracting the two and factoring yields Setting the first factor to zero yields Branch 1 already discussed. Setting the second factor to zero defines an ellipse when Ω < 5: This can be parameterized straightforwardly as Plugging this back into equation (2.4a) to solve for Ω and simplifying yields The solution satisfies the necessary condition that Ω < 5 for two subintervals of [0, π], yielding two additional branches: The branch diverges as θ approaches the endpoints of the interval. At θ = π 2 , this yields a = b = √ −Ω = 5 2 , i.e. it lies on Branch 1. This is the transcritical bifurcation. Setting the second factor of equation (2.3) to zero allows us to solve for a and c in the same manner that we solved for a and b in deriving Branches 3 and 4, Substituting these values back into equation (2.2a) or (2.2c) and simplifying yields a formula for b Finally, plugging these values into equation (2.2b) yields a quartic equation for Ω. This quartic equation factors into linear and cubic terms, so that Ω satisfies either Here we have defined µ = sin 2 θ, so that µ = 3 4 corresponds to θ = π 3 or θ = 2π 3 . • Branch 5 corresponds to solutions that make the linear factor zero. This yields physically meaningful solutions only for π 3 < θ < 2π 3 . At θ = π 2 , the branch touches Branch 4, and in fact corresponds to a symmetry-breaking branch for that mode.
Cardano's formula yields three solutions that make the cubic term equal zero. One of these is real for all θ and yields two branches of physically relevant solutions.
(Ω, Q) → (−∞, ∞) as the endpoints of the interval are approached. It arises in a saddlenode bifurcation at Ω ≈ −2.7. The other two roots of the cubic are not physically meaningful, with Ω > 1 or (Ω) = 0 for all values of θ. Fig. 2.2 shows all the branches discussed above. We summarize the conclusions of the parameterization study shown in this figure. Branches 1, 4, and 6 correspond to the ones discussed in Ref. [27] and are colored as in Fig. 1.3. Branch 1, the constant solution, begins at the origin and bifurcates in the (symmetry-breaking) pitchfork bifurcation at Ω = − 1 2 , creating Branch 6. Branch 1 then undergoes a transcritical bifurcation at Ω = − 5 2 , creating Branch 4 of symmetric solutions. Both halves of Branch 4 diverge to (−∞, ∞) in the (Ω, Q) plane. As Ω → −∞, b → 0 and a = c → +∞. The half branch that extends to the right from the bifurcation turns around at a fold bifurcation at Ω ≈ −1.94 (this value is the root of a quartic polynomial).
Branch 5 emerges from a symmetry breaking pitchfork bifurcation of Branch 4. For large negative Ω, the solution on Branch 5 concentrates at the center point b. Branch 7, which arises in a saddle-node bifurcation has no analog in Fig. 1.3, but we find many such branches in our more complete numerical study in Section 5 Branch 5 appears to cross Branch 6 in the (Ω, Q)-projection Ω ≈ −2.97, as the two branches appear to cross in Fig. 1.3. The statement has no bearing on the conclusions of Refs. [27,28]. The asymptotics of the solutions on the quantum graph for large L 2 norm depend on the exponential decay of the standing waves as a function of the distance from their maxima. The slope of the curve of symmetric solutions is steeper than that of the curve of symmetric solutions because the symmetric solution concentrates at the single vertex v 3 for large Q, whereas the asymmetric solution concentrates at the two points v 1 and v 2 for large values of Q.

Investigating the two bifurcations on the dumbbell quantum graph.
We now return to the dumbbell quantum graph. We present more general information before returning to the particular bifurcation problem.  This observation is made systematic using the secular determinant, which is welldescribed by Berkolaiko [8]. The nonzero eigenvalues are of the form λ = k 2 where k solves Σ(k) = det (I − SD(k)) = 0, (3.2) and the dimension of the square matrices S and D(k) is twice the number of edges. This equation is derived by assuming that the solution is given as a linear combination of e ikxj and e −ikxj on each edge e j , and enforcing that the vertex conditions be satisfied. For the dumbbell graph, After some manipulations, equation (3.2) for the dumbbell graph is equivalent to The three factors of equation ( If L is chosen as an integer multiple of π/2, eigenvalues from the loops may coincide with those of the even and odd eigenfunctions. We will assume no such resonances exist. Numerical examples, discretized using second-order finite differences, with ghost-points to implement the boundary conditions, are shown in Fig. 3.1. Yang's results apply to a more general nonlinearity G(Φ, x) = F (|Φ| 2 , x)Φ which here represents the cubic term in equation (1.9) and any terms that would arise due to a nonzero V (x) in the definition (1.6a). Solutions (Φ 0 , Λ 0 ) such that the operator (3.4) define the bifurcation points and L 10 will from here on denote the linearized operator at a bifurcation point. Assume that L 10 has a one-dimensional nullspace with null eigenfunction Υ. Define Then the result depends on the following quantities (3.7) While L 10 is by assumption not invertible, all the above terms involving L −1 10 are well-defined where they appear in the theorem below.
Yang's main result is the following.
These bifurcations are all essentially one-dimensional. The Lyapunov-Schmidt theorem, or, alternatively, center manifold theory, essentially allow the bifurcation problem to be reduced to an equation involving one variable a (due to the onedimensional null space of L 10 ) [22]. Following Ref. [19], bifurcations in a onedimensional problem depend on a normal form equation with an expansion of the form with the constants Θ j determining the type of the bifurcation.
Remark 3.1. From this, it is apparent that the transcritical bifurcation is more generic than the pitchfork bifurcation, in that it requires the vanishing of fewer terms in this expansion. Yang notes that if F (|Φ| 2 , x) = F (|Φ| 2 , −x), and if Φ 0 (x) is symmetric (even or odd) and Υ has the opposite symmetry, then Θ 1 = Θ 2 = 0, and pitchfork bifurcations are to be expected [39,Remark 2]. He constructed transcritical bifurcations in a numerical example by tuning the system to satisfy Θ 1 = 0, but only for potentials without this symmetry [40]. By contrast, the dumbbell quantum graph is symmetric (see the next section), yet certain symmetric solutions undergo transcritical bifurcations. We discuss the conditions giving rise to transcritical bifurcations further in Section 4.
Further, when a system has such a symmetry, then the pitchfork bifurcation is a symmetry-breaking bifurcation. That is, the system has a solution Φ that is invariant under symmetry transformation R(Φ) = Φ, whereas at the bifurcation, the new pair of solutions arises with Φ 1 = R(Φ 2 ) and Φ 2 = R(Φ 1 ) instead.

Remark 3.2.
A second theorem of Yang is useful for identifying bifurcation types from a graph of a standing wave's power Q versus its frequency Λ [39]. It shows that in a transcritical bifurcation, the two branches of solutions appear tangent to each other, while in a symmetry-breaking pitchfork bifurcation, the two branches overlap in the (Λ, Q) plot, and branch out from the point (Λ 0 , Q 0 ) at a nonzero angle. See the second row of Fig. 1.1. This has been seen in pitchfork bifurcations in NLS systems before, e.g. [29,34].
Remark 3.3. The pitchfork bifurcation is generic in systems possessing certain symmetries. When those symmetries are broken by a small general perturbation, the coefficient Θ 1 will become nonzero. Thus, a saddle-node bifurcation must occur, as is shown in Fig. 3.2(a). If the system is such that perturbations leave Θ 1 ≡ 0, but results in nonzero Θ 2 , then the pitchfork bifurcation splits into a saddle-node and a transcritical as shown in Fig. 3.2(b), with the two bifurcation points merging to a pitchfork in the limit Θ 2 → 0.

Validation by numerical continuation.
Ref. [27] finds that the constantvalued solution (1.10) to system (1.6) undergoes two bifurcations as Λ is decreased from zero. We numerically compute both branches using pseudo-arclength continuation. We use centered differences to discretize system (1.9), and second-order onesided differences to discretize the vertex condition (1.6c) on the dumbbell-shaped graph depicted in Fig. 1.2. The system is solved using Newton's method, combined with pseudo-arclength continuation as described in the textbook of Nayfeh and Balachandran [30]. The code has two advantages over that used in Ref. [27]: first, it is able to go around fold bifurcations, and second, it detects branching bifurcations, both pitchfork and transcritical, and is able to switch branches; following Govaerts [23]. The nonlinear solver uses a Newton conjugate gradient-based scheme developed by Yang [38].
The result of a computation with L = 2 is shown in Fig. 3.3. The symmetrybreaking bifurcation is exactly as in Fig. 1.3. In addition, this computation captures several features not present in [27], but visible in the bifurcation diagram for the bowtie DST model, Fig. 2.2. First, the branch of centered solutions of Fig. 1.3 undergoes a fold bifurcation at Λ ≈ −0.19. The upper branch then meets the branch of constant solutions at the "symmetry-preserving" bifurcation Fig. 1.3 near Λ = −ω 2 1 ≈ −0.215, in exactly the same manner found for DST on the bowtie graph. By comparison with Fig. 1.1, it appears that this should be a transcritical bifurcation, which will be confirmed below. In addition, the branch undergoes a symmetry-breaking pitchfork bifurcation very close to the saddle-node bifurcation. Solutions from both the top and bottom halves of the symmetric branch, as well as the symmetry-broken branch are displayed in Fig. 3.4.    The constants in equation (3.7) depend on L −1 10 Φ 0 . Since L 10 Φ| Φ=1 = −γ 2 , we have All of these quantities depends on known functions and quantities, except for Θ 5 , which depends on L −1 10Φ 2 and is only required at pitchfork bifurcations. In particular, at the pitchfork bifurcationΦ is an odd function, and thus the null space of L 10 consists entirely of constant multiples ofΦ, which are also odd. Thus, sinceΦ 2 is even, it is orthogonal to the null space of L 10 (which is self adjoint), and we can (non-uniquely) define L −1 3.4.1. The pitchfork bifurcation. Setting γ = Ω 1 , where Ω 2 1 is the eigenvalue corresponding to the first odd eigenfunction, then a bifurcation occurs at Λ = −Ω 2 1 /2. We call this eigenfunction Φ odd and normalize it such that Γ Φ 2 odd dx = 1. Then since Φ 3 odd (x) is odd, Θ 2 = 0. Since Θ 3 is positive and Θ 5 will in general be nonzero (though this would need to be checked numerically), the bifurcation is of pitchfork type.

ROY H. GOODMAN
Following [28], the perturbation expansion is given in terms of a generic small parameter a (compare with expansion (3.8)) by In order to solve for terms at O(a 2 ), we must satisfy a Fredholm condition at O(a 3 ) but we do not otherwise need the terms at that higher order. We find whereΦ 2 is the unique even-symmetric solution to L 10Φ2 = Φ 2 odd . This is well-defined since the null space of the self-adjoint operator on the left is spanned by the odd-symmetric function Φ odd and the right-hand side has even symmetry. Ref. [27] notes that this equation may be solved explicitly, and uses the exact solution to prove the direction in which the solution branch bends.

3.4.2.
The transcritical bifurcation. Setting γ = ω 1 , where ω 2 1 is the eigenvalue corresponding to the first even eigenfunction, then a bifurcation occurs at Λ = −ω 2 1 /2. We call this eigenfunction Φ even and normalize it such that Γ Φ 2 even dx = 1. Then since Φ 3 even (x) is even, then in general Θ 2 = 0. In this case, following [39], the perturbation solution is given by This leads a sequence of equations: O(a) : O(a 2 ) : At O(a) we find that Φ 1 = Φ even + C and β 1 = −2ω 1 C, with C to be determined. The equation at O(a 2 ) then becomes The constant term is solvable, but the remaining term must be orthogonal to Φ even for solvability. This then yields a condition Solving the equation at this order yields whereΦ 2 is the unique solution to L 10Φ2 = ω 1 4CΦ even + 3Φ 2 even ; Φ 2 , Φ even = 0.
A Fredholm condition at O(a 3 ) then yields Remark 3.4. The expansions described in this section are equally valid for every eigenvalue/eigenfunction pair in the even or odd families. Thus, the constant-valued solution undergoes a pitchfork bifurcation each time its value crosses a frequency −ω 2 j /2 and a transcritical bifurcation each time it crosses a frequency −Ω 2 j /2. We delay discussion of the bifurcations that involve the loop-localized families of eigenfunctions to Section 5.
Remark 3.5. Fig. 3.3 shows that that a saddle node bifurcation occurs very near to this transcritical. Remark 3.3 discusses the fact that a pitchfork bifurcation perturbed via a perturbation with Θ 1 = 0 splits into a saddle-node and a transcritical bifurcation. In the next section, we discuss a specific sense in which the dumbbell graph can be thought of as a perturbation of a simpler graph which features only pitchfork bifurcations. 4. Comparison with simpler quantum graphs. We were able to gain some insight into the dumbbell graph by considering the simpler problem of the bowtie combinatorial graph. Here we consider two additional simpler problems, the cubic Schrödinger problem defined on an interval with Neumann boundary conditions, and the problem on the lollipop graph, obtained from the graph in Fig. 1.2 by removing the edge e 3 and imposing a Neumann condition at vertex v 2 .
The former can be thought of as a quantum graph consisting of just one edge and two vertices. This system, which can be derived as a singular limit of the dumbbell graph, has a different bifurcation structure. A rigorous approach to quantum graphs in the limit of vanishing edge length is given by Berkolaiko et al. [9]. In what follows, we rescale the x coordinates along the edges of the dumbbell graph so that 1 = 3 = and 2 = π. The secular determinant equation (3.3) becomes: The solutions to these the three factors correspond, respectively in the order presented, to the even, odd, and loop-localized families of eigenfunctions. For the loop-localized families, this gives k = 2nπ/ , a sort of boundary layer, in that the solution is concentrated into a small region near the boundary of the → 0 limiting problem. For the other two branches we may expand the wavenumber k = ∞ j=0 j k j . For the even-symmetric solutions we find k 0 = 2n and for the odd-symmetric, k 0 = 2n − 1 and in both cases Letting the parameterization on the edge e 2 be given by 0 < x 2 < π, then all the eigenfunctions are of the form cos kx with k ≈ 2n on the even family and k ≈ (2n−1) on the odd family. Now, we take the singular limit → 0 + . The boundary layer solutions disappear, and the problem on the central edge e 2 reduces to the Schrödinger equation on an interval with Neumann conditions, whose eigenfunctions are simply cos nx. We examine the sequence of bifurcations undergone by the nonlinear standing wave with constant spatial profile Φ and frequency Λ = −2Φ 2 for this system. This solution undergoes a sequence of bifurcations at the discrete sequence of frequencies Λ n = −n 2 /2. Each of these bifurcations is a pitchfork, independently of the parity of n and in contrast to the dumbbell system, which alternates between pitchfork and transcritical bifurcations. In terms of Theorem 1, as lim →0 + Θ 2 = 0 and the transcritical becomes a pitchfork.
The difference between the problem on the interval and that on the dumbbell is that the Neumann problem on the interval has an additional symmetry: namely, the solution on the interval can be extended to the whole real line under an even periodic extension. Consider a solution in a neighborhood of the pitchfork bifurcation. In both the Neumann problem and the dumbbell problem, the solution looks to leading order like Φ = constant+aΦ odd . Reversing the sign of a is equivalent to applying the symmetry R 2 to this solution. This argument fails near the transcritical bifurcation on the dumbbell: reversing the sign of a is not equivalent to applying a symmetry. However, in the limit = 0, a new symmetry appears, as the evenly extended eigenfunctions have an odd symmetry about each of their zeros. This symmetry persists for nonlinear standing waves.
It is not necessary that there be loops appended to the interval. In the family of graphs pictured in Fig. 4.1, the extra symmetry of the Neumann interval is also destroyed, and pitchforks involving even eigenfunctions are be converted to transcriticals. All the calculations presented here and in [27] go through without alteration. In general, for any compact graph with Kirchhoff conditions at all vertices, the constant-valued solution will undergo a sequence of transcritical bifurcations, since the eigenfunctions of the linearized system will be the same as the linear eigenfunction and all are orthogonal to the constant-valued solution, yielding Θ 1 = 0. If, however, the graph is sufficiently symmetric that it supports odd eigenfunctions, then these will satisfy Θ 2 = 0 and the bifurcations will be symmetry-breaking pitchforks.
Therefore, we can expect that as 0, each of the transcritical bifurcations on the dumbbell graph becomes increasingly pitchfork-like. In other words, the transcritical bifurcation and the saddle-node nearby merge with each other, and Θ 2 → 0 at the bifurcation point. While this statement should hold in a neighborhood of the bifurcations, for large values of |Λ|, the two branches (corresponding to branches (a) and (b) in Figures 3.3 and 3.4) will contain, respectively, one or two pulses, and thus diverge from each other. We investigate this possibility by re-computing the continuation calculation shown in Fig. 3.3 for large values of L in Fig. 4.2, which correspond to small values of . These show the width of the branch narrows greatly for large L, and appears more like a pitchfork. Finally, we note that in the lollipop graph, lacking the symmetry of the dumbbell, the analysis leading to the pitchfork bifurcation in Section 3.4.1 is not applicable, but that the analysis in Section 3.4.2 would go through without any changes. Therefore we see that eliminating both loops leads to only pitchforks, while eliminating only one leads to only transcriticals. This is an answer to what is special about the dumbbell structure that allows both behaviors. The bifurcation diagram for the lollipop graph will become important in Section 5.4 when we discuss standing waves on the dumbbell that are built from lollipop standing waves.

5.
Numerical enumeration of stationary solutions. Given the attention that we have just paid to just two bifurcations, we should point out that the behavior of standing waves is significantly more complex. We have already pointed out that the constant-valued solution undergoes an alternating sequence of pitchfork and transcritical bifurcations, each time its frequency crosses Λ = −Ω 2 j /2 or Λ = −ω 2 j /2. In addition, each of the other linear eigenfunctions can be continued into the nonlinear regime in a similar manner. In the remainder of this section, we will enumerate all possible branches of stationary solutions using a combination of numerical continuation, phase planes, and analytical reasoning.
The restriction of equation (1.9) to any edge e j is just whose phase plane is shown in Fig. 5.1, with trajectories lying along level sets of the energy For Λ > 0, there is one family of periodic orbits surrounding a single elliptic fixed point at the origin. For Λ < 0, this fixed point has undergone a symmetry-breaking bifurcation, becoming hyperbolic and giving rise to two new fixed points. These are defined by equation (1.10), and define the constant-valued solutions to system (1.6). Exact formulas for the periodic orbits are provided in terms of Jacobi elliptic functions in Appendix A. A similar enumeration of standing waves was carried out for the tadpole graph by Noja et al. [33], and for more general graphs by Gnutzmann and Waltner [20,21]. Both groups have approached this by using the formulas for the exact solutions on the edges, Jacobi elliptic functions on finite edges and the sech function on infinite edges. Each of these partial solutions depends on undetermined parameters: e.g. the parameters κ or k, and τ in equations (A.1) and (A.4). These then must be chosen to satisfy the vertex conditions (1.6b) and (1.6c). Together, these determine a system of equations. The authors then use normal form perturbation theory to simplify this system of equations into a more tractable form, finding good agreement. 5.1. Two types of loops. We may classify the different types of stationary solutions by their behavior on the self-connected edge e 1 (or equivalently e 3 ). Since the solution along edge e 1 lies along a level set of the energy, continuity condition (1.6b) requires φ 1 (−π) = φ 1 (π) and thus, by the symmetry of the phase planes across the y-axis in We will call such a loop incomplete. Note that when L is not a multiple of π/2 both loops of eigenfunctions from the even and odd families are of this type; compare Fig. 3.1(a-d). At the vertex φ 1 = φ 2 , by the continuity condition (1.6b). Therefore when e 1 is an incomplete loop, any jump in the energy (5.2) at the vertex is determined by the 1 2 (φ j ) 2 term and equation (5.4) implies E(e 1 ) < E(e 2 ).
If Equation ( 1(e-f). Note that selfdirected edges on which the solution is identically zero can be considered both complete and incomplete. Since φ 1 (π) may be nonzero, condition (5.5) implies that when e 2 is a complete loop, then E(e 1 ) ≥ E(e 2 ). Therefore, there are three possible types of stationary solution. On the first, both loops are incomplete, on the second, both loops are complete, and the third contains one complete and one incomplete loop.

5.2.
Two incomplete loops. We can search for these stationary solutions with two incomplete loops using the following shooting argument. All such stationary solutions must satisfy dφj dxj xj =0 = 0, j = 1, 3. Now define f (q, Λ, L) = φ 3 (0) found by solving the following three consecutive initial value problems for equation (1.9): 1. Solve for φ 1 (x) from x 1 = 0 to x 1 = π with initial condition φ 1 (0) = q, . Then solutions of f (q, Λ, L) = 0 correspond to solutions of (1.9) on Γ with the appropriate vertex conditions. This definition is closely related to the map defined by Pelinovsky and Schneider over one cell of an infinite periodic graph [35]. We may then continue these solutions as a function of Λ to follow the branches and find bifurcations.
To follow this procedure, we first fix Λ = −1 and L = 2 and compute f (q, Λ, L) over an appropriate interval, which is shown in Fig. 5.2. By standard arguments about continuous dependence on initial conditions, f is a continuous function of q and we can guarantee finding all such solutions by resolving this graph adequately. We then use MATCONT, a MATLAB continuation package, to trace these branches, varying Λ and holding L fixed [14,15]. This is summarized in an incomplete but complex bifurcation diagram in Fig. 5.3, which follows all the solutions to f (q, −1, 2) = 0 with 0 < q < 1.3.  First, we find the previously mentioned branches arising alternately in pitchfork and transcritical bifurcations from the constant state. We also note that all other even-symmetric branches undergo symmetry breaking as well, as was previously seen in Fig. 3.3. This black curve is the centered branch found in Ref. [27]. Many additional families bifurcate into existence in saddle-node bifurcations. Some of these are plotted twice in Fig. 5.3(b), as they correspond to asymmetric solutions and their value at the center of either loop may be used as the input parameter q in the shooting function. An alternative to a numerical shooting method is to use the fact that the solution on each edge is given by a cnoidal or dnoidal function, see Appendix A, or by a hyperbolic secant and use the boundary conditions (1.6b) and (1.6c), together with the complete loop assumption to derive nonlinear equations for the parameters in these solutions. This is the approach taken in reference [33]. 5.3. Two complete loops. Solutions of this type can be completely described analytically, including the values at Λ for which they bifurcate. Doing so, however, is somewhat unwieldy, as there are many cases to consider. The restriction to each loop must be an integer number of periodic orbits of Equation (1.9), and on the restriction to central edge must be an integer number of half-period orbits, with φ 2 (±L) = 0. One such solution is shown in Fig. 5.4.
In Appendix A, it is shown that when Λ > 0, equation (5.1) has periodic orbits of period T < T max cn given by (A.2), and that for Λ < 0 it has cnoidal periodic orbits of all periods outside the separatrix, while inside the separatrix, it has dnoidal periodic orbits with period T > T min dn given by (A.5). We can classify all such solutions with a triple (n 1 , m, n 3 ) as follows  Therefore, the trivial solution (0, 0, 0) exists for all Λ, and the other solutions emerge as Λ is decreased. We present a minimal list, and omit additional solutions obtained from group orbits of these solutions under (1.12). • At Λ = n 2 1 , the solutions (n 1 , 0, n 1 ) and (n 1 , 0, 0) bifurcate from (0, 0, 0). Simultaneously, for all n 3 > n 1 , the solution (n 1 , 0, n 3 ) bifurcates from (0, 0, n 3 ).
• At Λ = m 2 Λ * , the solution (n 1 , m, n 3 ) bifurcates from (n 1 , 0, n 3 ) if m < 2Ln 1,3 /π. • At Λ = 0, the solution (Λ, Λ, Λ) bifurcates from (0, 0, 0). Simultaneously (n 1 , Λ, n 3 ) bifurcates from (n 1 , 0, n 3 ) for all n 1,3 ≥ 0. • At Λ = −m 2 Λ * /2, solutions of the form (n 1 , − |m| , n 3 ) bifurcate from those of the form (n 1 , Λ, n 3 ). Note that if m < 0 is odd and n 1 = n 3 then (n 1 , m, n 3 ) and (n 3 , m, n 1 ) are not related by symmetry.    A bifurcation diagram for these solutions on the dumbbell is shown in Fig. 5.7(b). Note that none of the solutions bifurcate from Q = 0. Points where the curve appears to end are actually saddle-node bifurcations corresponding to the crossings described in the previous paragraph. 6. Conclusion. To this author, reference [27] raised the question of whether the bifurcation scenario seen for NLS on the dumbbell graph is fundamentally different from that seen in other simpler systems. The far simpler bowtie DST system possesses the bifurcation structure identical to what they found, indicating that the phenomenon does not depend on the system being a quantum graph, but suggesting that the topology of the graph is important. This is confirmed by explicit comparison with the analogous problem posed on a line segment. The self-connected edges can be thought of as providing a singular perturbation to this simpler system, one which breaks the symmetry of half the families of solutions and thereby transforms pitchfork bifurcations into transcriticals. By contrast, the lollipop graph has less symmetry than the dumbbell and thus only transcriticals.
In the opposite direction, conceptually, we have shown how to enumerate the complete set of standing waves for this system, revealing a tremendous amount of complexity. Nonetheless, we have only been able to understand this much because of the relative simplicity of the dumbbell graph. In particular, the loop structure only permits two types of behavior at the vertices, and from this we are able to classify the solutions into three types. Further, the shooting methods described above depend strongly on the simple arrangement of the three edges. On a graph with many branches, or cycles, such shooting methods would be untenable.
We have not discussed stability of these standing waves as was done in [33], other than for the constant valued solution. Of special interest would be the stability of the solutions of the type (n 1 , 0, n 3 ) discussed in Section 5.2. Because the phase of the solution is ill-defined on the edge e 2 , the components on the other two edges can each be multiplied by an arbitrary phase. We may ask whether the stability depends on this phase. Beyond this, we can easily construct solutions which vanish on e 2 and for which the solutions on the other two edges oscillate at different frequencies.
Determining the stability of these solutions involves analyzing a genuinely timedependent problem requiring Floquet theory.
Finally, In addition, recent work by Kirr has made progress toward analysis that identifies all the coherent states of a given nonlinear wave system using global bifurcation theory and perturbations from the large-amplitude limit [25]. It would be of interest to see if such an approach could be useful in studying the Λ → −∞ limit of the standing wave problem. explaining the parameterization method to him. The author gratefully acknowledges the support of the Institute for Mathematics and its Applications (IMA) during the 2016-17 program in Mathematics and Optics, where this research was initiated. Mathematica was used for symbolic calculation and MATLAB for numerical calculation [36,37]. For Λ < 0, there are two types of periodic orbits. The first lie outside the separatrices shown in Fig. 5.1 and are given by the same formulas as above, only for 1 √ 2 < κ < 1. As κ → 1 − , these orbits approach the separatrices and their period diverges. As κ → 1 As k → 0 + , these periodic orbits shrink toward the nonzero fixed points and As k → 1 − , these orbits approach the separatrices and their periods diverge.