Nonlinear instability of half-solitons on star graphs

Kairzhan, Adilbek; Pelinovsky, Dmitry E.

doi:10.1016/j.jde.2018.02.020

Cited by 27 publications

(47 citation statements)

References 35 publications

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“…Remark 2.7. The result of Theorem 2.6 is very similar to the instability result in Theorem 2.7 in [14] which was proven for the uniform star graph with α = 1.…”

Section: Resultssupporting

confidence: 79%

“…When the center of mass for the shifted state reaches the vertex, the shifted state becomes a saddle point of energy under the fixed mass. At this point in time, orbital instability of the shifted state develops as a result of the saddle point geometry similar to the instability studied in [14].…”

Section: Introductionmentioning

confidence: 56%

“…It was proved in [15] (see Theorem 4.1 and Corollary 4.3) that the one-parameter family of shifted states in Lemma 2.1 is spectrally unstable for a < 0 and spectrally stable for a > 0. The degenerate state at a = 0 called the half-soliton state is expected to be nonlinearly unstable as was shown in [14] for uniform star graphs with α = 1. In regards to the shifted states with a > 0, it was conjectured in [15] (see Conjectures 7.1 and 7.2) that the shifted state in Lemma 2.1 with a > 0 is nonlinearly unstable under the time flow because the shifted states drifts along the only incoming edge towards the half-soliton state with a = 0, where it is affected by spectral instability of the shifted states with a < 0.…”

Section: Resultsmentioning

confidence: 84%

“…Similarly to Lemma 5.4 in [14], the following lemma gives coercivity of the quadratic forms associated with the operators L + (ω) and L − (ω).…”

Section: 2mentioning

confidence: 88%

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Drift of Spectrally Stable Shifted States on Star Graphs

Kairzhan¹,

Pelinovsky²,

Goodman³

2019

SIAM J. Appl. Dyn. Syst.

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When the coefficients of the cubic terms match the coefficients in the boundary conditions at a vertex of a star graph and satisfy a certain constraint, the nonlinear Schrödinger (NLS) equation on the star graph can be transformed to the NLS equation on a real line. Such balanced star graphs have appeared in the context of reflectionless transmission of solitary waves. Steady states on such balanced star graphs can be translated along the edges with a translational parameter and are referred to as the shifted states. When the star graph has exactly one incoming edge and several outgoing edges, the steady states are spectrally stable if their monotonic tails are located on the outgoing edges. These spectrally stable states are degenerate minimizers of the action functional with the degeneracy due to the translational symmetry. Nonlinear stability of these spectrally stable states has been an open problem up to now. In this paper, we prove that these spectrally stable states are nonlinearly unstable because of the irreversible drift along the incoming edge towards the vertex of the star graph. When the shifted states reach the vertex as a result of the drift, they become saddle points of the action functional, in which case the nonlinear instability leads to their destruction. In addition to rigorous mathematical results, we use numerical simulations to illustrate the drift instability and destruction of the shifted states on the balanced star graph.

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“…Remark 2.7. The result of Theorem 2.6 is very similar to the instability result in Theorem 2.7 in [14] which was proven for the uniform star graph with α = 1.…”

Section: Resultssupporting

confidence: 79%

Section: Introductionmentioning

confidence: 56%

Section: Resultsmentioning

confidence: 84%

“…Similarly to Lemma 5.4 in [14], the following lemma gives coercivity of the quadratic forms associated with the operators L + (ω) and L − (ω).…”

Section: 2mentioning

confidence: 88%

See 2 more Smart Citations

Drift of Spectrally Stable Shifted States on Star Graphs

Kairzhan¹,

Pelinovsky²,

Goodman³

2019

SIAM J. Appl. Dyn. Syst.

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“…Remark 6.2. For a more complete discussion of the instability of the half-soliton e iωtũ ω (x) as a solution to (6.6), including a calculation of Mor(L + ) by other means, we refer the reader to [12].…”

Section: (69)mentioning

confidence: 99%

The Maslov and Morse indices for Sturm-Liouville systems on the half-line

Howard¹,

Sukhtayev²

2020

Discrete &Amp; Continuous Dynamical Systems - A

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We show that for Sturm-Liouville Systems on the half-line [0, ∞), the Morse index can be expressed in terms of the Maslov index and an additional term associated with the boundary conditions at x = 0. Relations are given both for the case in which the target Lagrangian subspace is associated with the space of L 2 ((0, ∞), C n ) solutions to the Sturm-Liouville System, and the case when the target Lagrangian subspace is associated with the space of solutions satisfying the boundary conditions at x = 0. In the former case, a formula of Hörmander's is used to show that the target space can be replaced with the Dirichlet space, along with additional explicit terms. We illustrate our theory by applying it to an eigenvalue problem that arises when the nonlinear Schrödinger equation on a star graph is linearized about a half-soliton solution.(A2) We assume that P , V , and Q all approach well-defined asymptotic endstates at exponential rate. That is, we assume there exist self-adjoint matrices P + , V + , Q + ∈ C n×n , with P + , Q + positive definite, and constants C and η > 0 so thatand similarly for V (x) and Q(x). In addition, we assume |P ′ (x)| ≤ Ce −η|x| for a.e. x ∈ (0, ∞).(A3) For the boundary conditions, we take α 1 , α 2 ∈ C n×n , and for notational convenience, set α = (α 1 α 2 ). We assume rank α = n, αJα * = 0, which is equivalent to self-adjointness in this case. Here, J denotes the standard symplectic matrix J = 0 −I n I n 0 , with I n denoting the usual n × n identity matrix. We can think of (1.1) in terms of the operatorwith which we associate the domainand the inner productWith this choice of domain and inner product, L is densely defined, closed, and self-adjoint, so σ(L) ⊂ R.Our particular interest lies in counting the number of negative eigenvalues of L (i.e., the Morse index). We proceed by relating the Morse index to the Maslov index, which is described in Section 3. We find that the Morse index can be computed in terms of the Maslov index and an additional term associated with the boundary condition at x = 0.As a starting point, we define what we will mean by a Lagrangian subspace of C 2n . For comments about working in C 2n rather than R 2n , the reader is referred to Remark 1.1 of [10], and the references mentioned in that remark.Definition 1.1. We say ℓ ⊂ C 2n is a Lagrangian subspace of C 2n if ℓ has dimension n and (Ju, v) C 2n = 0, (1.3)for all u, v ∈ ℓ. Here, (·, ·) C 2n denotes the standard inner product on C 2n . In addition, we denote by Λ(n) the collection of all Lagrangian subspaces of C 2n , and we will refer to this as the Lagrangian Grassmannian.

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$$L^2$$-critical NLS on noncompact metric graphs with localized nonlinearity: topological and metric features

Dovetta

Tentarelli

2019

Calc. Var.

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Nonlinear instability of half-solitons on star graphs

Cited by 27 publications

References 35 publications

Drift of Spectrally Stable Shifted States on Star Graphs

Drift of Spectrally Stable Shifted States on Star Graphs

The Maslov and Morse indices for Sturm-Liouville systems on the half-line

$$L^2$$-critical NLS on noncompact metric graphs with localized nonlinearity: topological and metric features

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