Standing waves of the quintic NLS equation on the tadpole graph

The tadpole graph consists of a circle and a half-line attached at a vertex. We analyze standing waves of the nonlinear Schr\"{o}dinger equation with quintic power nonlinearity equipped with the Neumann-Kirchhoff boundary conditions at the vertex. The profile of the standing wave with the frequency $\omega\in (-\infty,0)$ is characterized as a global minimizer of the quadratic part of energy constrained to the unit sphere in $L^6$. The set of minimizers includes the set of ground states of the system, which are the global minimizers of the energy at constant mass ($L^2$-norm), but it is actually wider. While ground states exist only for a certain interval of masses, the standing waves exist for every $\omega \in (-\infty,0)$ and correspond to a bigger interval of masses. It is shown that there exist critical frequencies $\omega_0$ and $\omega_1$ such that the standing waves are the ground states for $\omega \in [\omega_0,0)$, local minimizers of the energy at constant mass for $\omega \in (\omega_1,\omega_0)$, and saddle points of the energy at constant mass for $\omega \in (-\infty,\omega_1)$. Proofs make use of both the variational methods and the analytical theory for differential equations.


Introduction
The analysis of nonlinear PDEs on metric graphs has recently attracted a certain attention [29]. One of the reason is potential applicability of this analysis to physical models such as Bose-Einstein condensates trapped in narrow potentials with T-junctions or X-junctions, or networks of optical fibers. Another reason is the possibility to rigorously prove a complicated behavior of the standing waves due to the interplay between geometry and nonlinearity, which is hardly accessible in higher dimensional problems.
The most studied nonlinear PDE on a metric graph G is the focusing nonlinear Schrödinger (NLS) equation with power nonlinearity, which we take in the following form: where the wave function Ψ(t, ·) is defined componentwise on edges of the graph G subject to suitable boundary conditions at vertices of the graph G. The Laplace operator ∆ and the power nonlinearity are also defined componentwise. The natural Neumann-Kirchhoff boundary conditions are typically added at the vertices to ensure that ∆ is self-adjoint in L 2 (G) with a dense domain D(∆) ⊂ L 2 (G) [12,20]. The Cauchy problem for the NLS equation (1.1) is locally well-posed in the energy space H 1 C (G) := H 1 (G) ∩ C 0 (G), which is the space of the componentwise H 1 functions that are continuous across the vertices of the graph G [15,22,24]. The following two conserved quantities of the NLS equation ( Date: January 6, 2020. and the total energy . Due to conservation of mass and energy and the Gagliardo-Nirenberg inequality (for which see [3,4]), unique local solutions to the NLS equation (1.1) in H 1 C (G) are extended globally in time for subcritical (p < 2) nonlinearities and for the critical (p = 2) nonlinearity in the case of small initial data.
Standing waves of the NLS equation (1.1) are solutions of the form Ψ(t, x) = e iωt Φ(x), where Φ satisfies the elliptic system (1.4) − ∆Φ − (p + 1)|Φ| 2p Φ = ωΦ and ω ∈ R is a real parameter. We refer to ω as the frequency of the standing wave and to Φ as to the spatial profile of the standing wave. The stationary NLS equation (1.4) is the Euler-Lagrange equation for the augmented energy functional or simply the action which is defined for every U ∈ H 1 C (G). If the infimum of the constrained minimization problem: is finite and is attained at Φ ∈ H 1 C (G) so that E µ = E(Φ) and µ = Q(Φ), we say that this Φ is the ground state. By the usual bootstrapping arguments, the same Φ is also a strong solution Φ ∈ H 2 NK (G) to the stationary NLS equation (1.4) with the corresponding Lagrange multiplier ω which depends on the mass µ. Here H 2 NK (G) is the space of the componentwise H 2 functions that satisfy the natural Neumann-Kirchhoff boundary conditions across the vertices of the graph G. This space coincides with the domain D(∆) of the self-adjoint Laplace operator ∆.
Ground states on metric graphs with Neumann-Kirchhoff boundary conditions at the vertices of G and no external potentials exist under rather restrictive topological conditions (see [3,4,5,6]). When delta-impurities at the vertices or external potentials give rise to a negative eigenvalue of the linearized operator at the zero solution, a ground state always exist in the subcritical [15], critical [13], and supercritical [11] cases.
The tadpole graph T has been proven to be a good testing ground for a more general study. A first classification of standing waves for the cubic (p = 1) NLS on the tadpole graph was given in [14], then it was extended to the subcritical case p ∈ (0, 2) in [30] where orbital stability of some standing waves has been considered.
By Theorem 2.2 in [3] for the subcritical case p ∈ (0, 2), E µ in (1.6) satisfies the bounds where E R + is the energy of a half-soliton of the NLS equation on a half-line with the same mass µ and E R is the energy of a full soliton on a full line with the same mass µ. By Theorem 3.3 and Corollary 3.4 in [4], the infimum is attained if there exists Ψ * ∈ H 2 NK (G) such that E(Ψ * ) ≤ E R . Based on this criterion, it was shown in [4] that the subcritical NLS equation for the tadpole graph T admits the ground state Φ for all positive values of the mass µ. Moreover, by using suitable symmetric rearrangements it was shown in [4] that the ground state Φ is given by a monotone piece of soliton on the half-line glued with a piece of a periodic function on the circle, with a single maximum sitting at the antipodal point to the vertex (see also [14,30]).
In the critical power p = 2, it was shown in [5,Theorem 3.3] that the ground state on the metric graph G with exactly one half-line (e.g., on the tadpole graph T ) is attained if and only if µ ∈ (µ R + , µ R ], where µ R + is the mass of the half-soliton of the NLS equation on the half-line and µ R is the mass of the full-soliton on the full line, both values are independent on ω for p = 2. Indeed, let ϕ ω (x) = |ω| 1/4 sech 1/2 (2 |ω|x) be a soliton of the quintic NLS equation on the line centered at x = 0, then we compute Thus, the ground state on the tadpole graph T exists if and only if µ ∈ (µ R + , µ R ]; moreover, E µ < 0. It was also shown in [5,Proposition 2.4] We conjecture that this behavior of E µ for the critical NLS equation (1.1) with p = 2 is associated to the decay of strong solutions Ψ(t, ·) ∈ H 2 NK (T ) to zero as t → ∞ if the mass µ of the initial data Ψ(0, ·) = Ψ 0 satisfies µ ≤ µ R + and the blow-up in a finite time t if µ > µ R . The latter behavior is known on the full line [16] but it has not been proven yet in the context of the unbounded metric graph T (strong instability of bound states on star graphs was recently analyzed in [22]).
The main novelty of this paper is to explore the variational methods and the analytical theory for differential equations in order to construct the standing waves with profile Φ satisfying the elliptic system (1.4) with p = 2, rewritten again as The variational construction relies on the following constrained minimization problem: (1.14) . We are not aware of previous applications of the variational problem (1.14) in the context of the NLS equation on metric graphs.
Versions of the variational problem (1.14) arise in the determination of the best constant of the Sobolev inequality, which is equivalent to the Gagliardo-Nirenberg inequality in R n (see, for example, [7,8,18,28] and references therein). However, as follows from [5] and it is shown in Lemma 2.5 below, the minimizer of (1.14) does not give the best constant in the Gagliardo-Nirenberg inequality on the tadpole graph T . The variational problem (1.14) gives generally a larger set of standing waves compared to the set of ground states in the variational problem (1.6). This turns out to be relevant for the orbital stability of the standing waves.
Another well-known variational problem is the minimization of the action functional (1.5) on the Nehari manifold. This approach was was used in [21] for the so-called delta potential on the line and generalized in [2] in the context of a star graph with a delta potential at the vertex. More recently, the variational problems at the Nehari manifolds were exploited in [31,9]. In Appendix B, we show how the constrained minimization problem (1.14) is related to the minimization of the action (1.5) on the Nehari manifold defined by the constraint B ω (U) = 3 U 6 L 6 (T ) . We shall now present the main results of this paper. The first theorem states that the variational problem (1.14) is useful to generate a family of standing waves Φ(·, ω) to the elliptic system (1.13) for every ω < 0. Theorem 1. For every ω < 0, there exists a global minimizer Ψ(·, ω) ∈ H 1 C (T ) of the constrained minimization problem (1.14), which yields a strong solution Φ(·, ω) ∈ H 2 NK (T ) to the stationary NLS equation (1.13). The standing wave Φ is real up to the phase rotation, positive up to the sign choice, symmetric on [−π, π] and monotonically decreasing on [0, π] and [0, ∞).
The main idea in the proof of Theorem 1 is a compactness argument which eliminates the possibility that the minimizing sequence splits or escapes to infinity along the unbounded edge of the tadpole graph T .
In what follows, we usually omit the dependence on ω for Ψ(·, ω) and Φ(·, ω). The linearization of the stationary NLS equation (1.13) around Φ is defined by the self-adjoint operator L : H 2 NK (T ) ⊂ L 2 (T ) → L 2 (T ) given by the following differential expression: Since it is self-adjoint, the spectrum of L in L 2 (T ) is a subset of real line. Since Φ(x) → 0 as x → ∞ exponentially on the half-line, application of Weyl's Theorem yields that the absolute continuous part of the spectrum of L is given by and that there are only finitely many eigenvalues of L located below |ω| with each eigenvalue having finite multiplicity. Let n(L) be the Morse index (the number of negative eigenvalues of L with the account of their multiplicities) and z(L) be the nullity index of the kernel of L (the multiplicity of the zero eigenvalue of L). Since (1.18) LΦ, Φ L 2 (Γ N ) = −12 Φ 6 L 6 (T ) < 0, there is always a negative eigenvalue of L so that n(L) ≥ 1. Since Φ is obtained from the variational problem (1.14) with only one constraint, by Courant's Min-Max Theorem, we have n(L) ≤ 1, hence n(L) = 1. In addition, we prove that the operator L is nondegenerate for every ω < 0 with z(L) = 0. These facts are collected together in the following theorem. The proof of Theorem 2 relies on the dynamical system methods and the analytical theory for differential equations. In particular, we construct the standing wave of Theorem 1 by using orbits of a conservative system on a phase plane and by introducing the period function, whose analytical properties are useful to prove monotonicity of parametrization of the standing wave in Lemma 3.1 and the non-degeneracy of the linearized operator L in Lemma 3.2.
In connection with the variational characterization of the standing waves on metric graphs T which are not necessarily ground states we mention two recent papers treating situations different than ours. In [32], local and not global constrained minimizers of the energy at constant mass in the critical power p = 2 are discussed for some cases of unbounded graphs with Neumann-Kirchhoff boundary conditions. In [6], local minima of the energy at constant mass for subcritical power are constructed for general graphs by means of a variational problem with two constraints.
The present paper is organized as follows. Section 2 gives the proof of Theorem 1 by using the variational characterization of the standing waves. Section 3 gives the proof of Theorem 2 by using the dynamical system methods and the analytical theory for differential equations. Section 4 gives the proof of Theorem 3 with computations of the asymptotical properties of the map ω → µ(ω) as ω → 0 and ω → −∞. Appendix A gives the precise characterization of the spectrum of the Laplace operator ∆ on the tadpole graph T . Appendix B gives information between the variational problem (1.14) and the minimization of the action (1.5) at the Nehari manifold. Appendix C gives computational details of approximating of the integral for the mass µ(ω) in the limit ω → −∞.

Variational characterization of the standing waves
Here we shall prove Theorem 1. First, we show that there exists a global minimizer Ψ ∈ H 1 C (T ) of the variational problem (1.14) for every ω < 0. Then, we deduce properties of the minimizer and use the Lagrange multipliers to obtain the solution Φ ∈ H 2 NK (T ) to the stationary NLS equation (1.13). We begin to recall that it is well known that problem 1.14 has a solution on both R and R + (see for example the already mentioned papers [7,8] or references therein). The precise value of the infima B R (ω) and B R + (ω) are given in the subsequent formula (2.11). Now let us consider the tadpole graph.
. Hence, it follows that B ω (U) ≥ 0 so that the infimum B(ω) > 0 of the variational problem (1.14) exists, where positivity of B(ω) follows from the nonzero constraint U L 6 (T ) = 1 and Sobolev's embedding of H 1 (T ) to L 6 (T ) in the sense that there exists a U-independent constant C > 0 such that for all U ∈ H 1 C (T ). Let {U n } n∈N be a minimizing sequence in H 1 (T ) such that U n L 6 (T ) = 1 for every n ∈ N and B ω (U n ) → B(ω) as n → ∞. Therefore, there exists a weak limit of the sequence in H 1 (T ) denoted by U * so that By Fatou's Lemma, we have L 6 (T ) = 1, so that γ ∈ [0, 1]. The following two lemmas eliminate the cases γ ∈ (0, 1) and γ = 0. Lemma 2.1. For every ω < 0, either γ = 0 or γ = 1. If γ = 1, then U * ∈ H 1 (T ) is a global minimizer of (1.14).
Proof. By Lemma 2.1, either γ = 0 or γ = 1, so we only need to exclude the case γ = 0. Let us define the variational problem analogous to (1.14) but posed on the line: . As is well known, the infimum B R (ω) is attained at the scaled soliton λ R ϕ ω satisfying the constraint λ R ϕ ω 6 L 6 (R) = 1, from which it follows that Evaluating the integrals in (2.7) yields the exact expression: We first show that if γ = 0, then the minimizing sequence {U n } n∈N escapes to infinity along the half-line in T as n → ∞ so that U * = 0 and B(ω) ≥ B R (ω). Then, we show that for every ω < 0 there exists a trial function U 0 ∈ H 1 C (T ) such that U 0 L 6 (T ) = 1 and B ω (U 0 ) < B R (ω). Therefore, the minimizing sequence cannot escape to infinity so that γ = 0. Hence γ = 1 by Lemma 2.1.
To proceed with the first step, let {U n } n∈N be a minimizing sequence such that U n ∈ H 1 C (T ), U n L 6 (T ) = 1, lim n→∞ B ω (U n ) = B(ω) and suppose that U n → 0 weakly in H 1 (T ). For simplicity, we consider the nonnegative sequence with U n ≥ 0. Let ǫ n ≥ 0 be the maximum of U n on [−π, π] ∪ [0, 2π] ⊂ T . Since U n → 0 as n → ∞ uniformly on any compact subset of T , we have ǫ n → 0 as n → ∞. Let us defineŨ n = (ũ n ,ṽ n ) ∈ H 1 C (T ) from the components of U n = (u n , v n ) as follows: . By the proof of Proposition B.1 in Appendix B, minimizing B ω (U) under the constraint U L 6 (T ) = 1 is the same as minimizing B ω (U) in U L 6 (T ) ≥ 1, therefore, {Ũ n } n∈N is also a minimizing sequence for the same variational problem (1.14). At the same time, the image ofŨ n covers all values in (0, max x∈T U n (x)) at least twice. IfŨ s n is the symmetric rearrangement ofŨ n on the line R, then it follows from the Polya-Szegö inequality on graphs (see Proposition 3.1 in [3]), that , n ∈ N. By taking the limit n → ∞, we obtain B(ω) ≥ B R (ω) in the case of γ = 0.
To proceed with the second step, we construct a trial function U 0 ∈ H 1 C (T ) such that U 0 L 6 (T ) = 1 and B ω (U 0 ) < B R (ω) with the following explicit computation. For every ω < 0, we define x ∈ (0, ∞), hence U 0 on T is a scaled soliton λ 0 ϕ ω truncated on [−π, ∞). Then, λ 0 is found from the normalization condition: It is clear that f (0) = 1 and lim A→∞ f (A) = 2 2/3 . We shall prove that f (A) < 2 2/3 for every A > 0. Indeed, for every A > 0, , z := sinh(A). Since Remark 2.3. Any smooth and compactly supported function in H 1 (R) can be considered as an element of H 1 C (T ) (see the proof of Theorem 2.2 of [3] and Remark 2.2 in [5]), so that by a density argument we have (independently on γ) Hence, for the escaping minimizing sequence with γ = 0 we would actually have that B(ω) = B R (ω). However, the existence of the trial function U 0 ∈ H 1 C (T ) such that U 0 L 6 (T ) = 1 and B ω (U 0 ) < B R (ω) eliminates the case γ = 0.
It follows from Lemmas 2.1 and 2.2 that there exists a global minimizer Ψ ∈ H 1 C (T ) of the variational problem (1.14) for every ω < 0. We now verify properties of the global Proof. If Ψ is a minimizer of the variational problem (1.14), so is |Ψ|. Hence we may assume that Ψ is real and positive. To prove symmetry and monotonic decay, we observe that if the minimizer is not symmetric on [−π, π] and is not decreasing on [0, π] and [0, ∞), then it is possible to define a suitable competitor on the tadpole graph T with lower value of B(ω), by using the well known technique of the symmetric rearrangements and the Polya-Szegö inequality on graphs (see Proposition 3.1 in [3], examples discussed after Corollary 3.4 in [4] and in [19]).
The following lemma gives as further information with a more precise quantitative control of the infimum B(ω). This result is similar to the energy bounds in (1.10) obtained for the subcritical nonlinearity.
Lemma 2.5. For every ω < 0, the infimum B(ω) in (1.14) satisfies the bounds Proof. The upper bound in (2.10) is verified in the proof of Lemma 2.2. In order to prove the lower bound, we use the Gagliardo-Nirenberg inequality on graphs: By Theorem 3.3 in [5], it follows that K T = K R + and the constant K R + is attained by the half soliton λ R + ϕ ω normalized by λ R + ϕ ω L 6 (R + ) = 1. By similar computations as in the proof of Lemma 2.2, we obtain that from which it also follows that By using the definition and the value of K T , we obtain for every U ∈ H 1 C (T ), where the latter inequality follows from the minimization of f (x) : Proof. By using Lagrange multipliers in Σ ω,ν (U) := B ω (U) − ν U 6 L 6 (T ) , we obtain Euler-Lagrange equation for Ψ: Since H 1 C (T ) is a Banach algebra with respect to multiplication, |Ψ| 4 Ψ ∈ H 1 C (T ), so that one can rewrite the Euler-Lagrange equation (2.12)

Dynamical system methods for the standing waves
Here we shall prove Theorem 2. We do so by using the dynamical system methods for characterization of the standing wave Φ ∈ H 2 NK (T ) of Theorem 1. In particular, we reduce the stationary NLS equation (1.13) to the second-order differential equation on an interval, for which we introduce the period function. By using the analytical theory for differential equations, we show monotonicity of the period function, which allows us to control nullity of the linearization operator L in (1.16).
Let Φ ∈ H 2 NK (T ) be a real and positive solution to the stationary NLS equation (1.13) with ω < 0 constructed by Theorem 1. For every ω < 0, we set ω = −ε 4 and introduce the scaling transformation for Φ = (u, v) as follows: The boundary-value problem for (U, V ) is rewritten in the component form: By the symmetry property in Theorem 1, we have U(−z) = U(z), z ∈ [−πε 2 , πε 2 ]. By uniqueness of the soliton ϕ on the half-line up to the spatial translation, we have V (z) = ϕ(z + a), z ∈ [0, ∞) for some a ∈ R, where ϕ(z) = sech 1/2 (2z). By the monotonicity property in Theorem 1, we have a ∈ (0, ∞). These simplifications allow us to reduce the existence problem (3.2) to the simplified form where a ∈ (0, ∞) and ε ∈ (0, ∞).   (Figure 2). By the boundary conditions in the system (3.3), the solution is related to the monotonically decreasing part of the homoclinic orbit shown by red line on the phase plane (U, U ′ ) in such a way that the value of U at the vertex is continuous, where the value of U ′ at the vertex jumps by half of its value. The green line is an image of the homoclinic orbit after U ′ is reduced by half. The value of U at the vertex is adjusted depending on the value of ε in the length of the interval [0, πε 2 ].
The phase plane representation of the solution to the boundary-value problem (3.2) shown on Fig. 2 is made rigorous in the following lemma. Proof. The differential equation −U ′′ + U − 3U 5 = 0 can be solved in quadrature with the first-order invariant: Since the value of E is constant in z, we obtain the exact value of E for the admissible solution to the system (3.3): Let U 0 (a) := ϕ(a), so that the map (0, ∞) ∋ a → U 0 (a) ∈ (0, 1) is C 1 and monotonically decreasing. Let U + (a) be the largest positive root of E + U 2 − U 6 = 0 such that U + (a) ≥ U * := 1 3 1/4 , which exists if E ∈ (E 0 , 0), where E 0 := − 2 3 √ 3 . It follows from (3.5) that the latter requirement is satisfied for every a ∈ (0, ∞). By means of the first-order invariant (3.4), the boundary-value problem (3.3) is solved in the following quadrature: Since E, U 0 , and U + are uniquely defined by a ∈ (0, ∞), the value of ε is uniquely defined by (3.6) from the value of a ∈ (0, ∞). It remains to prove that the map (0, 1) ∋ U 0 → ε(U 0 ) ∈ (0, ∞) is C 1 and monotonically decreasing. Since the map (0, ∞) ∋ a → U 0 (a) ∈ (0, 1) is C 1 and monotonically decreasing, the two results imply that the composite map the map (0, ∞) ∋ a → ε(a) ∈ (0, ∞) is C 1 and monotonically increasing, which yields the assertion of the lemma with the solution U given in the implicit form by the quadrature (3.6).
To prove monotonicity of the C 1 map (0, 1) ∋ U 0 → ε(U 0 ) ∈ (0, ∞), we use the technique developed for the flower graph in the cubic NLS equation [26]. We define the following period function: where A(u) = u 2 − u 6 , U + is the largest positive root of E + A(u) = 0 such that U + ≥ U * := 1 3 1/4 , and E is given by E = 1 4 A(U 0 ) − A(U 0 ) = − 3 4 A(U 0 ). The value U * is the only critical (maximum) point of A(U) on R + with A(U * ) = 2 Define v := E + A(u) and compute for every u ∈ (0, 1): All terms in this expression are non-singular for every u ∈ (0, 1). It enables us to express the period function T (U 0 ) in the equivalent way: where we have used that 2 E + A(U 0 ) = A(U 0 ). The right-hand side is C 1 in U 0 on (0, 1), which proves that the map (0, 1) Moreover, we compute the derivative explicitly by where we have used that dE dU 0 = − 3 4 A ′ (U 0 ). We need to prove that T ′ (U 0 ) < 0. Note that E + A(U * ) > 0 and the last term in the right-hand side of (3.7) is negative. In order to analyze the first term in the right-hand side of (3.7), we compute directly If U 0 ∈ (U * , 1), then A ′ (U 0 ) < 0 and the right-hand side of (3.8) is negative for every u ∈ [U 0 , U + ]. Hence the first term in the right-hand side of (3.7) is negative and so is In order to study T ′ (U 0 ) for U 0 ∈ (0, U * ), for which A ′ (U 0 ) > 0, we need to integrate the expression in (3.8) by parts: where the first term is negative and the last term is positive. Recall that A ′ (U 0 ) > 0 for U 0 ∈ (0, U * ). Combining the last negative term in the right-hand side of (3.7) and the first positive term obtained from the previous expression yields which is negative for every U 0 ∈ (0, U * ). Hence the right-hand side of (3.7) is negative and so is T ′ (U 0 ) for every U 0 ∈ (0, U * ). Thus, we have proven that T ′ (U 0 ) < 0 for every U 0 ∈ (0, 1), from which it follows that the map (0, 1) ∋ U 0 → ε(U 0 ) ∈ (0, ∞) is monotonically decreasing.
We consider the most general solution of LΥ = 0 and prove that Υ / ∈ H 2 N K (T ) for every ω < 0. We use the representation ω = −ε 4 and the scaling transformation (3.1) for Φ = (u, v) ∈ H 2 NK (T ). Similarly, we represent Υ = (u, v) by using the scaling transformation from which the following boundary-value problem is obtained for (U, V): We are looking for a solution (U, V) ∈ H 2 N K (T ) to the boundary-value problem (3.10) so that V(z) → 0 as z → ∞. Recall that V (z) = ϕ(z + a) with a ∈ (0, ∞) defined uniquely in terms of ε ∈ (0, ∞). Then, the only decaying solution to the second equation in the system (3.10) takes the form: where α ∈ C is arbitrary. The general solution to the first equation in the system (3.10) can be written in the form: where β, γ ∈ C are arbitrary and W is a linearly independent solution to U ′ . Thanks to the symmetry of the coefficients to the first equation in the system (3.10), W (−z) = W (z) and U ′ (−z) = −U ′ (z). By using the boundary conditions in the system (3.10), we obtain the linear system on coefficients of the solutions (3.11) and (3.12): Since U ′ (πε 2 ) = 1 2 ϕ ′ (a) = 0 for every a ∈ (0, ∞), it follows from the system (3.13) that β = 0 and a nonzero solution for (α, γ) exists if and only if (3.14) W (πε 2 ) = 0 and We shall now express the even solution W to −U ′′ + U − 15U 4 U = 0. Let U(z; E) be an even solution of the first-order invariant (3.4) with free parameter E < 0 normalized by the boundary condition U(0; E) = U + (E), where U + (E) is the largest positive root of E + U 2 − U 6 = 0 such that U + (E) ≥ U * := 1 3 1/4 . Let E(ε) be defined for every ε > 0 by the boundary conditions: where a ∈ (0, ∞) is uniquely defined from ε ∈ (0, ∞) by Lemma 3.1.
The assertion of Theorem 2 is proven.
It remains to consider the asymptotic limits of µ(ω) as ω → 0 and ω → −∞. This will be done separately with the use of two different asymptotic methods.
The limit ω → 0 is handled by using the power series expansions.
Lemma 4.1. For small ω < 0, we have Proof. The limit ω → 0 corresponds to the limit ε → 0, for which solutions of the boundary-value problem (3.3) can be obtained by power series: where U 0 = U(0). From the boundary conditions in (3.3) we obtain and Combining these expansions yields nonlinear equations for U 0 and a in terms of ε, which are uniquely solved with the following power expansion: and (4.4) a = 2πε 2 − 28π 3 ε 6 + O(ε 10 ).
The limit ω → −∞ is handled by using properties of elliptic functions.
We shall now explore the asymptotic limit ε → ∞, which corresponds to the limit a → ∞. It follows from (3.5) in the limit a → ∞ that By solving the cubic equation for |ρ 3 | in (4.10) and using the explicit expressions for ρ 1,2 , we obtain in the same limit: , from which we obtain Approximations of elliptic functions in terms of hyperbolic functions (see 16.15 in [1]) were justified in Proposition 4.6 and Appendix A in [27]. In the limit k → 1 and  It follows from (4.11), (4.12), (4.13), and (4.14) that as long as 1 − k = O(e −2πε 2 ) and a = O(ε 2 ) as ε → ∞. Since By differentiating the exact solution (4.11) and using (4.12), (4.13), (4.15), and (4.16), we obtain Since the boundary conditions in In combination with the expansion for k in (4.13), this yields the unique asymptotic balance at or equivalently, This completes the asymptotic construction of the solution (4.11) in the limit ε → ∞.
We can now compute the mass µ(ω) given by (1.2) versus ε as ε → ∞. As it is explained in Appendix C, we obtain On the other hand, thanks to the asymptotic balance (4.18) we have Since ω = −ε 4 < 0, the asymptotic expansion (4.21) yields (4.8).
Let us illustrate numerically the implicit solution defined by the quadrature (3.6). For each fixed U 0 ∈ (0, 1), we find U + ∈ [U * , 1), where U * := 1 3 1/4 , from numerical solution of E + U 2 − U 6 = 0 with E given by (3.5). Then, we integrate the quadrature (3.6) numerically, hence obtaining a unique value of ε 2 = |ω| 1/2 for each U 0 . Then, we compute the mass µ from the following integral: where a ∈ (0, ∞) is expressed from U 0 ∈ (0, 1) by the explicit formula: By using the numerical integration above, we have obtained the mapping ω → µ(ω), which is plotted on Figure 1. Figure 3 shows the asymptotic dependencies (4.1) and (4.8) by solid lines superposed together with the numerical data for µ(ω) by black dots. The levels (1.11) and (1.12) are shown by dotted lines. The agreement between the asymptotic and numerical data is excellent. Proof. Let us consider the spectral problem −∆U = λU with U = (u, v) ∈ H 2 NK (T ). Due to the geometry of the tadpole graph T , the spectrum of −∆ in L 2 (T ) is the union of two sets: the set of λ for which v = 0 and the set of λ for which v = 0.
Eigenvalues of the spectral problem (A.1) are located at {0, 1, 4, 9, . . . } and for each λ = n 2 , n ∈ N, the eigenfunction of −∆ is given by The second set includes the absolute continuous spectrum of −∆ located on [0, ∞) and for each λ = k 2 with k ∈ [0, ∞) the Jost function of −∆ is given by It remains to check if the second set includes isolated eigenvalues λ < 0 with v = 0. Representing a possible eigenfunction of −∆ for λ < 0 as we obtain from the Neumann-Kirchhoff boundary conditions (1.7) that λ is a solution to the transcendental equation: (A.6) 1 + 2 tanh(π |λ|) = 0, which has no roots for real λ. B ω (U) − 3 U 6 L 6 (T ) = 0, which characterizes the set of solutions of the stationary NLS equation (1.13). The following proposition establishes a relation of the latter minimization problem with minimization of B ω (U) at fixed U 6 L 6 (T ) . Note that this result is not used in the main part of our paper and is added here for completeness.
Euler Appendix C. Asymptotic computation of the integral (4.20) Here we justify the asymptotic computation of the integral (4.20). By using the exact solution (4.11) and the asymptotic expansions (4.12) and (4.13) as a → ∞, we obtain I ε := it follows that I ε in (C.1) can be expanded as ε → ∞ in the form: