# Nonlinear Instability of Half-Solitons on Star Graphs

Adilbek Kairzhan Department of Mathematics, McMaster University, Hamilton, Ontario L8S 4K1, Canada  and  Dmitry Pelinovsky Department of Mathematics, McMaster University, Hamilton, Ontario L8S 4K1, Canada
December 31, 2020
###### Abstract.

We consider a half-soliton stationary state of the nonlinear Schrödinger equation with the power nonlinearity on a star graph consisting of edges and a single vertex. For the subcritical power nonlinearity, the half-soliton state is a degenerate critical point of the action functional under the mass constraint such that the second variation is nonnegative. By using normal forms, we prove that the degenerate critical point is a nonlinear saddle point, for which the small perturbations to the half-soliton state grow slowly in time resulting in the nonlinear instability of the half-soliton state. The result holds for any and arbitrary subcritical power nonlinearity. It gives a precise dynamical characterization of the previous result of Adami et al., where the half-soliton state was shown to be a saddle point of the action functional under the mass constraint for and for cubic nonlinearity.

This research was supported by the NSERC Discovery Grant.

## 1. Introduction

In many realistic physical experiments involving wave propagation in thin waveguides, multi-dimensional models can be reduced approximately to the one-dimensional PDEs on graphs [9, 14, 15, 20]. Similarly, quantum nanowires and other thin structures in nanotechnology can be described by the one-dimensional Schrödinger equation on graphs .

Spectral properties of Laplacian and other linear operators on graphs have been intensively studied in the past twenty year [10, 13]. The time evolution of linear PDEs on graphs is well defined by the standard semi-group theory, once a self-adjoint extension of the graph Laplacian is constructed. On the other hand, the time evolution of nonlinear PDEs on graphs is a more challenging problem involving interplay between nonlinear analysis, geometry, and the spectral theory of non-self-adjoint operators. The nonlinear PDEs on graphs, mostly the nonlinear Schrödinger equation (NLS), has been studied in the past five years in the context of existence, stability, and propagation of solitary waves .

In a series of papers [1, 2, 3, 4, 5], Adami, Cacciapuoti, Finco, and Noja analyzed variational properties of stationary states on a star graph, which is the union of half-lines (edges) connected at a single vertex. For the standard Kirchhoff boundary conditions at the vertex and for odd , there is only one stationary state of the NLS on the star graph. This state is represented by the half-solitons along each edge glued by their unique maxima at the vertex. By using a one-parameter deformation of the NLS energy constrained by the fixed mass, it was shown that the half-soliton state is a saddle point of the constrained NLS energy . On the other hand, by adding a focusing delta impurity to the vertex, it was proven that there exists a global minimizer of the constrained NLS energy for a sufficiently small mass below the critical mass [1, 3, 4]. This minimizer coincides with the -tail state symmetric under exchange of edges, which has monotonically decaying tails and which becomes the half-soliton state if the delta impurity vanishes. In the concluding paper , it was proven that although the constrained minimization problem admits no global minimizers for a sufficiently large mass above the critical mass, the -tail state symmetric under exchange of edges is still a local minimizer of the constrained NLS energy when a focusing delta impurity is added to the vertex.

Due to local minimization property, the -tail state symmetric under exchange of edges is orbitally stable in the time evolution of the NLS in the presence of the focusing delta impurity. Although the second variation of the constrained energy was mentioned in the first work , the authors obtained all the variational results in [3, 4, 5] from the energy formulation avoiding the linearization procedure. In the same way, the saddle point geometry of energy at the half-soliton state in the case of vanishing delta impurity was not related in  to the instability of the half-soliton state in the time evolution of the NLS. It is quite well known that the saddle point geometry does not necessarily imply instability of stationary states in Hamiltonian systems because of the presence of neutrally stable modes of negative energy .

The recent works of Adami, Serra, and Tilli [6, 7] were devoted to the existence of ground states on the unbounded graphs that are connected to infinity after removal of any edge. It was proven that the infimum of the constrained NLS energy on the unbounded graph coincides with the infimum of the constrained NLS energy on the infinite line and it is not achieved (that is, no ground state exists) for every such a graph with the exception of graphs isometric to the real line . The reason why the infimum is not achieved is a possibility to minimize the constrained NLS energy by a family of NLS solitary waves escaping to infinity along one edge of the graph. The star graph with vanishing delta impurity is an example of the unbounded graphs with no ground states, moreover, the constrained NLS energy of the half-soliton state is strictly greater than its infimum. Thus, the study in  provides a general argument of the computations in , where it is shown that the one-parameter deformation of the half-soliton state with the fixed mass reduces the NLS energy and connects the half-soliton state with the solitary wave escaping along one edge of the star graph.

Further works on existence and stability of stationary states on the unbounded graphs have been developed in the context of the logarithmic NLS equation , the power NLS equation with interactions , the power NLS equation on the tadpole graph , and the cubic NLS equation on the periodic graph [16, 27].

In the present work, we provide a dynamical characterization of the result in  for the NLS with the power nonlinearity and in the case of an arbitrary star graph. By using dynamical system methods (in particular, normal forms), we will verify that the half-soliton state is the saddle point of the constrained NLS energy on the star graph and moreover it is dynamically unstable due to the slow growth of perturbations. This nonlinear instability is likely to result in the destruction of the half-soliton state pinned to the vertex and the formation of a solitary wave escaping to infinity along one edge of the star graph.

Since the nonlinear saddle points are rarely met in applications of the NLS equations, it is the first time to the best of our knowledge when the energy method is adopted to the proof of the nonlinear instability of the stationary states.

The paper is organized as follows. Section 2 states the main results for the NLS equation on the star graph. Positivity of the second variation of the action functional is proven in Section 3. Saddle point geometry near the half-soliton state is proven with normal forms in Section 4. Dynamical characterization of the nonlinear instability of the half-soliton state is developed with normal forms and energy estimates in Section 5.

## 2. Main results

Let be a star graph, which is constructed by attaching half-lines at a common vertex. Let us choose the vertex as the origin and parameterize each edge of by . The Hilbert space on the graph is given by

 L2(Γ)=⊕Nj=1L2(R+).

Elements in are represented in the componentwise sense as vectors of -functions with each component corresponding to one edge. The squared norm of such -functions is given by

 ∥Ψ∥2L2(Γ):=N∑j=1∥ψj∥2L2(R+).

Similarly, we define the -based Sobolev spaces on the graph

 Hk(Γ)=⊕Nj=1Hk(R+),k∈N

and equip them with suitable boundary conditions at the vertex. For the weak formulation of the NLS on , we define by using the continuity boundary conditions as in

 (2.1) H1Γ:={Ψ∈H1(Γ):ψ1(0)=ψ2(0)=⋯=ψN(0)},

whereas for the strong formulation of the NLS on , we define by using the Kirchhoff boundary conditions as in

 (2.2) H2Γ:={Ψ∈H2(Γ):ψ1(0)=ψ2(0)=⋯=ψN(0),N∑j=1ψ′j(0)=0}.

The dual space to is .

We consider the NLS equation on the star graph with the power nonlinearity in the normalized form,

 (2.3) i∂Ψ∂t=−ΔΨ−(p+1)|Ψ|2pΨ,

where , , is the Laplacian operator defined in the componentwise sense with primes denoting derivatives in , and the nonlinear term is interpreted as a symbol for .

In the weak formulation, stationary states of the NLS are defined as critical points of the action functional in the energy space , where is a free parameter, whereas

 (2.4) E(Ψ)=∥Ψ′∥2L2(Γ)−∥Ψ∥2p+2L2p+2(Γ),Q(Ψ)=∥Ψ∥2L2(Γ)

are the energy and mass functionals, respectively. The local well-posedness of the NLS evolution in follows by a standard contraction method. The energy and mass are constants of motion in the time evolution of the NLS flow in . See Propositions 2.1 and 2.2 in .

###### Remark 2.1.

The local solutions of the NLS in are extended globally in time by the energy conservation and Gagliardo–Nirenberg inequality for the -subcritical power nonlinearity with . On the other hand, local solutions to the NLS are known to blow up in a finite time in the norm for (critical nonlinearity) and (supercritical nonlinearity).

In the strong formulation, stationary states of the NLS are given by the standing wave solutions of the form

 Ψ(t,x)=eiωtΦω(x),Φω∈H2Γ,

where are real-valued solutions of the stationary NLS equation,

 (2.5) −ΔΦω−(p+1)|Φω|2pΦω=−ωΦω,Φω∈H2Γ.

The weak and strong formulations of the stationary states of the NLS on are equivalent to each other because the Kirchhoff conditions in are natural boundary conditions for critical points of in .

No solution to the stationary NLS equation (2.5) exists for , because in and as if by Sobolev’s embedding theorems. Therefore, we consider in the stationary NLS equation (2.5). Since consists of edges with the parametrization on , the scaling transformation

 (2.6) Φω(x)=ω12pΦ(z),z=ω12x

can be used to scale the positive parameter to unity. The normalized profile is now a solution of the stationary NLS equation

 (2.7) −ΔΦ+Φ−(p+1)|Φ|2pΦ=0,Φ∈H2Γ.

The stationary NLS equation (2.7) has a particular solution

 (2.8) Φ(x)=ϕ(x)⎛⎜ ⎜ ⎜ ⎜⎝11⋮1⎞⎟ ⎟ ⎟ ⎟⎠\rm withϕ(x)=sech1p(px).

This solution is labeled as the half-soliton state. If is odd, the half-soliton state is a unique solution to the stationary NLS equation (2.7) in , whereas if is even, there exists additional solutions with the translational parameters. See Theorem 5 in . In what follows, we only consider the half-soliton state for any .

Our main results are given as follows. Thanks to the scaling transformation, we set and use notations and for and .

###### Theorem 2.2.

Let be the Hessian operator for the second variation of at in . For every , it is true that for every , where

 (2.9) L2c:={V∈L2(Γ):⟨V,Φ⟩L2(Γ)=0}.

Moreover, if and only if belongs to a -dimensional subspace of spanned by an orthogonal set . Consequently, is a degenerate minimizer of in .

###### Remark 2.3.

If , then if and only if belongs to a -dimensional subspace of with an additional degeneracy. For , the second variation is not positive in .

###### Theorem 2.4.

Let be defined in Theorem 2.2. For every , there exists such that for every satisfying , there exists a unique minimizer of the variational problem

 (2.10)

such that for a -independent constant . Moreover, is sign-indefinite in . Consequently, is a nonlinear saddle point of in with respect to perturbations in .

###### Remark 2.5.

The restriction is used in order to expand up to the cubic terms with respect to the perturbation and then to pass to normal forms. If , is still a nonlinear saddle point of in but the proof needs to be modified by the fact that is -dimensional. If , it follows already from the second derivative test that is a saddle point of in .

###### Theorem 2.6.

For every , there exists such that for every (sufficiently small) there exists with such that the unique global solution to the NLS equation (2.3) starting with the initial datum satisfies

 (2.11) infθ∈R∥e−iθΨ(t0)−Φ∥H1(Γ)>ϵ\rm for some \;t0>0.

Consequently, the orbit is unstable in the time evolution of the NLS equation (2.3) in .

###### Remark 2.7.

If , the instability claim of Theorem 2.6 follows from analysis of the NLS equation on the real line [11, 24]. If , the instability claim of Theorem 2.6 follows from the standard approach .

Theorems 2.2, 2.4, 2.6 are proven in Sections 3,4, and 5, respectively.

## 3. Proof of Theorem 2.2

We set by the scaling transformation (2.6) and consider the half-soliton state given by (2.8). Substituting with real-valued into the action functional and expanding in yield

 (3.1) Λ(Φ+U+iW)=Λ(Φ)+⟨L+U,U⟩L2(Γ)+⟨L−W,W⟩L2(Γ)+N(U,W),

where

 ⟨L+U,U⟩L2(Γ) := ∫Γ[(∇U)2+U2−(2p+1)(p+1)Φ2pU2]dx, ⟨L−W,W⟩L2(Γ) :=

and

 N(U,W)=⎧⎪⎨⎪⎩o(∥U+iW∥2H1(Γ)),p∈(0,12),O(∥U+iW∥3H1(Γ)),p≥12.

In the strong formulation, we can also define the two Hessian operators

 (3.2) L+ = −Δ+1−(2p+1)(p+1)Φ2p:H2Γ⊂L2(Γ)→L2(Γ), (3.3) L− = −Δ+1−(p+1)Φ2p:texttextH2Γ⊂L2(Γ)→L2(Γ),

where . Both operators are extended as the self-adjoint operators in . The spectrum consists of the continuous and discrete parts denoted by and respectively.

By Weyl’s Theorem, since is bounded and decays exponentially fast to zero at infinity, we have . Therefore, we are only concerned with the eigenvalues of in . The following result shows that , is simple, and is coercive in the subspace associated with a single constraint in (2.9).

###### Lemma 3.1.

There exists such that

 (3.4) ⟨L−W,W⟩L2(Γ)≥C∥W∥2H1(Γ)\rm for every \;W∈H1Γ∩L2c,

where is given by (2.9).

###### Proof.

By using (2.8), we write for every ,

 (3.5)

By using , , and integration by parts, we obtain

 ∫+∞0pw2jϕ2pdx=∫+∞02wjdwjdxϕ′ϕdx

and

 ∫+∞0(w2j−ϕ2pw2j)dx=∫+∞0w2j(ϕ′ϕ)2dx,

so that the representation (3.5) is formally equivalent to

 (3.6) ⟨L−W,W⟩L2(Γ)=N∑j=1∫+∞0ϕ2∣∣ddx(wjϕ)∣∣2dx≥0.

Since for every and , the representation (3.6) is justified for every . It follows from (3.6) that if and only if satisfies

 (3.7) ddx(wjϕ)=0\rm almost everywhere% and for every \;j.

Sobolev’s embedding of into and equation (3.7) imply that for some constant . The continuity boundary conditions in the definition of in (2.1) then yield , which means that is a simple eigenvalue of the operator in (3.3) with the eigenvector . Since eigenvalues of are isolated, the coercivity bound (3.4) follows by the spectral theorem and Gårding inequality. ∎

In order to study in , we recall the construction of exponentially decaying solutions of the second-order differential equation

 (3.8) −u′′(x)+u(x)−(2p+1)(p+1)sech2(px)u(x)=λu(x),x∈(0,∞),λ<1.

The following two lemmas recall some well-known results on the Schrödinger equation (3.8).

###### Lemma 3.2.

For every , there exists a unique solution to equation (3.8) such that

 (3.9) limx→+∞u(x)e√1−λx=1.

The other linearly independent solution to equation (3.8) diverges as .

###### Proof.

See, e.g., Lemma 5.2 in . The existence of a unique decaying solution as is obtained after the boundary–value problem (3.8)–(3.9) is reformulated as the Volterra’s integral equation with a bounded kernel. The other linearly independent solution to the second-order equation (3.8) diverges as thanks to the -independent and nonzero Wronskian determinant between the two solutions. ∎

###### Lemma 3.3.

If (resp. ) for some , the solution of Lemma 3.2 is extended into an odd (resp. even) eigenfunction of the Schrödinger equation (3.8) on the infinite line. The point becomes the eigenvalue of the associated Schrödinger operator defined in . There exists exactly one corresponding to and a simple eigenvalue corresponding to , all other such points in are bounded away from zero.

###### Proof.

The solution in Lemma 3.2 is extended to an eigenfunction of the associated Schrödinger operator defined in by the reversibility of the Schrödinger equation (3.8) with respect to the transformation . The count of eigenvalues follows by Sturm’s nodal theorem since the odd eigenfunction for the eigenvalue ,

 ϕ′(x)=−sech1p(px)tanh(px)

has one zero on the infinite line. Hence, is the second eigenvalue of the associated Schrödinger equation with exactly one simple negative eigenvalue that corresponds to an even eigenfunction. All other eigenvalues in are bounded away from zero. ∎

###### Remark 3.4.

For , the solution in Lemma 3.2 is available in the closed analytic form:

 u(x)=e−√1−λx3−λ+3√1−λtanhx−3sech2x3−λ+3√1−λ.

In this case, the eigenvalues and eigenfunctions in Lemma 3.3 are given by

 λ=−3: u(x)=14sech2x, λ=0: u(x)=12tanhxsechx.

No other eigenvalues of the associated Schrödinger operator on exist in .

By using Lemmas 3.2 and 3.3, we can now characterize in . The following result shows that includes a simple negative eigenvalue and a zero eigenvalue of multiplicity .

###### Lemma 3.5.

Let be a solution of Lemma 3.2 for . Then, is an eigenvalue of if and only if either or (both and cannot vanish simultaneously). Moreover, in has multiplicity if and multiplicity if .

###### Proof.

Let be an eigenvalue of and denote the eigenvector by . Since and decay to zero as , by Sobolev’s embedding of to the space , we can parameterize by using from Lemma 3.2 as follows

 U(x)=u(x)⎛⎜ ⎜ ⎜ ⎜⎝c1c2⋮cN⎞⎟ ⎟ ⎟ ⎟⎠,

where are some coefficients. By using the boundary conditions in the definition of in (2.2), we obtain a homogeneous linear system on the coefficients:

 (3.10) c1u(0)=c2u(0)=⋯=cNu(0),c1u′(0)+c2u′(0)+⋯+cNu′(0)=0.

The determinant of the associated matrix is

 (3.11) Δ=[u(0)]N−1u′(0)∣∣ ∣ ∣ ∣ ∣ ∣∣1−10…010−1…0100…0⋮⋮⋮⋱⋮111…1∣∣ ∣ ∣ ∣ ∣ ∣∣=N[u(0)]N−1u′(0).

Therefore, is an eigenvector for an eigenvalue if and only if , which is only possible in (3.11) if either or . Moreover, multiplicity of and in coincides with the multiplicity of the eigenvalue because it gives the number of linearly independent solutions of the homogeneous linear system (3.10). The assertion of the lemma is proven. ∎

###### Corollary 3.6.

There exists exactly one simple negative eigenvalue in and a zero eigenvalue in of multiplicity , all other possible eigenvalues of in are bounded away from zero.

###### Proof.

The result follows from Lemmas 3.3 and 3.5. ∎

###### Remark 3.7.

For the simple eigenvalue in , the corresponding eigenvector is

 U=u(x)⎛⎜ ⎜ ⎜ ⎜⎝11⋮1⎞⎟ ⎟ ⎟ ⎟⎠,

where for every with . For the eigenvalue of multiplicity in , the invariant subspace of can be spanned by an orthogonal basis of eigenvectors . The orthogonal basis of eigenvectors can be constructed by induction as follows:

 N=3: U(1)=ϕ′(x)⎛⎜⎝1−10⎞⎟⎠,U(2)=ϕ′(x)⎛⎜⎝11−2⎞⎟⎠, N=4: U(1)=ϕ′(x)⎛⎜ ⎜ ⎜⎝1−100⎞⎟ ⎟ ⎟⎠,U(2)=ϕ′(x)⎛⎜ ⎜ ⎜⎝11−20⎞⎟ ⎟ ⎟⎠,U(3)=ϕ′(x)⎛⎜ ⎜ ⎜⎝111−3⎞⎟ ⎟ ⎟⎠,

and so on.

The following result shows that the operator is positive in the subspace associated with a scalar constraint in (2.9), provided the nonlinearity power is in , and coercive on a subspace of orthogonal to .

###### Lemma 3.8.

For every , for every , where is given by (2.9). Moreover if and only if belongs to the -dimensional subspace in the kernel of . Consequently, there exists such that

 (3.12) ⟨L+U,U⟩L2(Γ)≥Cp∥U∥2H1(Γ)\rm for every \;U∈H1Γ∩L2c∩[Xc]⊥.
###### Proof.

Since by Weyl’s Theorem, the eigenvalues of at and given by Corollary 3.6 are isolated. Since for every , exists in and is in fact given by up to an addition of an arbitrary element in . By the well-known result (see Theorem 3.3 in ), (that is, restricted on subspace ) is nonnegative if and only if

 (3.13) 0≥⟨L−1+Φ,Φ⟩L2(Γ)=−⟨∂ωΦω|ω=1,Φ⟩L2(Γ)=−12ddω∥Φω∥2L2(Γ)∣∣∣ω=1.

Moreover, if . By the direct computation, we obtain

 ∥Φω∥2L2(Γ)=Nω1p−12∫∞0ϕ(z)2dz

so that

 (3.14) ddω∥Φω∥2L2(Γ)=N(1p−12)ω1p−32∫∞0ϕ(z)2dz,

so that if and if . This argument gives the first two assertions of the lemma. The coercivity bound (3.12) follows from the spectral theorem in and Gårding inequality. ∎

• Proof of Theorem 2.2. It follows from the expansion (3.1) that

 12⟨Λ′′(Φ)V,V⟩L2(Γ)=⟨L+U,U⟩L2(Γ)+⟨L−W,W⟩L2(Γ)\rm withV=U+iW,

where are real-valued. The result of Theorem 2.2 follows by Lemmas 3.1 and 3.8.

## 4. Proof of Theorem 2.4

To prove Theorem 2.4, it is sufficient to work with real-valued perturbations to the critical point of the action functional . Assuming , we substitute with real-valued into and expand in to obtain

 (4.1) Λ(Φ+U)=Λ(Φ)+⟨L+U,U⟩L2(Γ)−23p(p+1)(2p+1)⟨Φ2p−1U2,U⟩L2(Γ)+S(U),

where

 S(U)=⎧⎪⎨⎪⎩o(∥U∥3H1(Γ)),p∈(12,1),O(∥U∥4H1(Γ)),p≥1.

Compared to the expansion (3.1), we have set and have expanded the cubic term explicitly, under the additional assumption . In what follows, we inspect convexity of with respect to the small perturbation .

The quadratic form is associated with the same operator given by (3.2). By Lemma 3.8, for every , where the orthogonal vectors are constructed inductively in Remark 3.7. Furthermore, by Lemma 3.8, if , that is, if satisfies , then the quadratic form is positive for , whereas if , the quadratic form is coercive. Hence, we inspect positivity of and use the orthogonal decomposition for :

 (4.2) U=c1U(1)+c2U(2)+⋯+cN−1U(N−1)+U⊥,

where . Therefore, for every and the coefficients are found uniquely by

 cj=⟨U,U(j)⟩L2(Γ)∥U(j)∥2L2(Γ),\rm for every \;j.

The following result shows how to define a unique mapping from to for small .

###### Lemma 4.1.

For every , there exists and such that for every satisfying , there exists a unique minimizer of the variational problem

 (4.3) infU⊥∈H1Γ∩L2c∩[Xc]⊥[Λ(Φ+c1U(1)+c2U(2)+⋯+cN−1U(N−1)+U⊥)−Λ(Φ)].

satisfying

 (4.4) ∥U⊥∥H1(Γ)≤A∥c∥2.
###### Proof.

First, we find the critical point of with respect to for a given small . Therefore, we set up the Euler–Lagrange equation in the form , where

 (4.5) F(U⊥,c):X×RN−1↦Y,X:=H1Γ∩L2c∩[Xc]⊥,Y:=H−1Γ∩L2c∩[Xc]⊥

is given explicitly by

 F(U⊥,c):=L+U⊥−p(p+1)(2p+1)ΠcΦ2p−1(N−1∑j=1cjU(j)+U⊥)2−ΠcR(N−1∑j=1cjU(j)+U⊥),

where is the orthogonal projection operator and satisfies

 ∥R(U)∥H1(Γ)=⎧⎪⎨⎪⎩o(∥U∥2H1(Γ)),p∈(12,1),O(∥U∥3H1(Γ)),p≥1.

Operator function satisfies the conditions of the implicit function theorem:

• is a map from to ;

• ;

• is invertible with a bounded inverse from to .

By the implicit function theorem (see Theorem 4.E in ), there are and such that for each with there exists a unique solution of the operator equation with such that the map

 (4.6) RN−1∋c→U⊥(c)∈X

is near and . Since is strictly positive, the associated quadratic form is coercive according to the bound (3.12), hence the critical point is a unique infimum of the variational problem (4.3) near .

It remains to prove the bound (4.4). To show this, we note that

satisfies for a -independent constant . Since is a map from to and , we have , so that the map (4.6) satisfies the bound (4.4). ∎

• Proof of Theorem 2.4. Let us denote

 (4.7)

where the infimum is achieved by Lemma 4.1 for sufficiently small . Thanks to the representation (4.1) and the bound (4.4), we obtain , where

 (4.8) M0(c):=−23p(p+1)(2p+1)N−1∑i=1N−1∑j=1N−1∑k=1cicjck⟨Φ2p−1U(i)U(j),U(k)
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