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arxiv: 2507.10821 · v5 · submitted 2025-07-14 · 🧮 math.AP

Nonlinear Schr\"odinger Equations on looping-edge graphs with δ'-type interactions

Jaime Angulo Pava , Alexander Munoz This is my paper

Pith reviewed 2026-05-19 03:57 UTC · model grok-4.3

classification 🧮 math.AP
keywords nonlinear Schrödinger equationorbital stabilitystanding wavesδ' interactionslooping-edge graphsdnoidal solutionsgraph Laplacians
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The pith

Standing waves for the cubic nonlinear Schrödinger equation on looping-edge graphs with δ' interactions are orbitally stable under specific tail conditions.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper investigates standing-wave solutions to the cubic nonlinear Schrödinger equation on a graph made of a circle and N attached infinite half-lines. The authors construct these waves by placing Jacobian elliptic dnoidal profiles on the circle and matching them either to zero or to soliton tails on the half-lines, while respecting the δ' vertex conditions that require continuous derivatives but allow jumps in the wave function. For the case of trivial zero tails, they prove orbital stability holds for every nonzero value of the interaction parameter Z. In the nontrivial tail case, which requires Z negative, both existence and stability or instability are shown to depend on the number of half-lines, the value of Z, and the wave's phase velocity. A reader would care because these results extend known stability theory from simpler domains to network-like structures that model phenomena in quantum mechanics and nonlinear wave propagation.

Core claim

We consider the self-adjoint realization of the Laplacian with δ'-type vertex conditions on a graph consisting of a circle and N half-lines. On the circle we use Jacobian elliptic dnoidal profiles combined with either trivial or soliton tail profiles on the half-lines, with full derivative matching at the attachment point. For trivial tails orbital stability holds for all Z not equal to zero. For nontrivial tails with Z negative we establish existence and orbital (in)stability depending on the relative size of N, Z and the phase velocity.

What carries the argument

Matching of Jacobian elliptic dnoidal profiles on the circle to zero or soliton tails on half-lines that satisfy the δ' vertex conditions requiring derivative continuity without wave function continuity at the vertex.

If this is right

  • For the trivial tail case, orbital stability is established for all nonzero Z.
  • For the non-trivial tail case with Z negative, existence of standing waves is shown.
  • Orbital stability or instability in the non-trivial case depends on N, Z, and the phase velocity of the standing wave.
  • The δ' conditions allow for a broader class of solutions than standard δ conditions on graphs.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • If these stability results hold, they suggest that similar constructions could apply to other vertex conditions or more complex graph topologies like multiple loops.
  • Physically, this could model stable light propagation or quantum particle behavior on wire networks with specific junction rules.
  • Testing the boundary between stability and instability regions numerically for specific N and Z values could confirm the parameter dependence.

Load-bearing premise

The analysis assumes that Jacobian elliptic dnoidal profiles on the circle can be matched to either zero or soliton tails on the half-lines while satisfying the δ' vertex conditions and full derivative continuity at the attachment point.

What would settle it

A numerical or analytical counterexample where a proposed dnoidal profile fails to satisfy the δ' conditions or where a supposedly stable standing wave exhibits exponential growth under small perturbations would disprove the stability claims.

Figures

Figures reproduced from arXiv: 2507.10821 by Alexander Munoz, Jaime Angulo Pava.

Figure 1
Figure 1. Figure 1: A wave function U defined on G will be understood as an (N + 1)-tuple of functions U = (ϕ,(ψj ) N j=1) = (ϕ, ψ1, . . . , ψN ), where ϕ is defined on e0 = [−L, L] and ψj is defined on ej = [L, ∞) for j = 1, . . . , N. We discuss in this work the case of nonlinear Schr¨odinger-type equations (NLS) iUt + ∆U + |U| p−1U = 0, p > 1, (1.1) where the action of the Laplacian operator ∆ on a general graph G is given… view at source ↗
Figure 2
Figure 2. Figure 2: T -shaped metric star graph. Let us give some examples on the tadpole graph. Assume N = 1. With the next example we pretend to resemble the so-called dipole-interaction of strength τ = 1. Example 6. Let γ ∈ R \ {0}. Consider the space Y0 = span{(γ, 1 γ , 0)T }. Note U⃗ ∈ Y0 if and only if γ 2ϕ(−L) = ϕ(L). Also, QU⃗ ′ ∈ Y ⊥ if and only if ϕ ′ (−L) = γ 2ϕ ′ (L). We therefore have that D(HY0 ) = {u ∈ D(H∗ 0 )… view at source ↗
Figure 3
Figure 3. Figure 3: For a looping-edge graph with N = 3, we exhibit profile elements in D′ Z with ϕ as centered periodic dnoidal wave and trivial tail-type profiles on all half-lines. 6. Dynamics of dnoidal plus soliton profiles on a looping-edge graph In this section, we establish some applications of the results above for studying the existence and orbital stability of standing-wave solutions to (1.1) in the cubic case (p =… view at source ↗
Figure 4
Figure 4. Figure 4: Graph of the periodic function χ0(x) with k = 1 2 . that α = 0 and so f ≡ 0. For Z < 0 and ω ̸= N2 Z2 we get −ω ∈ ρ(Hδ ′ Z ) and so g ≡ 0. Therefore, the kernel of L+,Z is trivial. 2) We note that n(L+,Z) ⩾ 1. Indeed, since ⟨L+,Z(Φω, 0) t ,(Φω, 0)⟩ = −2 Z L −L Φ 3 ω dx < 0, the min-max principle implies that there is at least one negative eigenvalue. Now, let (f, g) ∈ D′ Z , with g = (gj ) N j=1, such that… view at source ↗
Figure 5
Figure 5. Figure 5: Plots of χ0 and Φw with sample translations with a = 0.6L and a = 0.9L using k = 1 2 . Theorem 6.9. Let us consider the self-adjoint operator (L−,Z, D′ Z ) defined in (6.13) for fixed Z, associated with the dnoidal profile (Φω, 0) given in Proposition 6.1 for ω > π 2 2L2 , namely, L−,Z = diag − ∂ 2 x + ω − Φ 2 ω, −∂ 2 x + ω, . . . , −∂ 2 x + ω  . (6.28) Then, the following hold: (1) The kernel of L−,Z is … view at source ↗
Figure 6
Figure 6. Figure 6: Orange: homoclinic orbit for the standing NLS equation for positive solutions. Green: Soliton tail to be considered. Red: the line v = N Z u. Blue: intersection points of the periodic orbit at the derivative level of q. Purple: inner periodic orbit starting and ending at the same qi . Purple + Green: a feasible profile Θ ∈ D′ Z . 6.3. Stability of dnoidal+tail soliton profiles on looping edge graphs. In th… view at source ↗
Figure 7
Figure 7. Figure 7: For N = 3, a sketch of a profile in D′ Z with Z < 0 where Φω,a is a shifted periodic dnoidal wave and ψ1 = ψ2 = ψ3 are tail-type profiles. it follows that n(L+,Z) ≧ 1. Moreover, from (6.4) it follows that L−,Z  Φω,a (Ψω,Z,N ) N i=1 =  0 0  and L+,Z  Φ ′ ω,a (Ψ′ ω,Z,N ) N i=1 =  0 0  . Note that Φ ′ ω,a,(Ψ′ ω,Z,N ) N i=1 ∈/ D′ Z . Indeed, suppose instead that Φ ′ ω,a,(Ψ′ ω,Z,N ) N i=1 ∈ D′ Z . In … view at source ↗
read the original abstract

In this work, we study the existence and orbital (in)stability of certain standing-wave solutions for the cubic nonlinear Schr\"odinger equation (NLS) posed on a looping-edge graph $\mathcal{G}$, consisting of a circle and a finite number $N$ of infinite half-lines attached to a common vertex. We consider the self-adjoint realization $(\mathcal{H}_Z, D(\mathcal{H}_Z))$ of the Laplacian, where the domain $D(\mathcal{H}_Z)$ encodes on the half-lines a $\delta'$-type vertex conditions (continuity of derivatives at the vertex, without requiring continuity of the wave function) and $Z \in \mathbb{R}\setminus\{0\}$. On the circle, we propose Jacobian elliptic profiles of dnoidal type combined with either trivial (zero) or soliton tail profiles on the half-lines with full derivative matching at the boundary. For the trivial tail case we establish orbital stability for all $Z \in \mathbb{R}\setminus\{0\}$, while for the non-trivial tail case (which requires $Z < 0$) we establish both existence and orbital (in)stability depending on the relative size of $N$, $Z$, and the phase velocity of the standing wave.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript constructs standing-wave solutions to the cubic NLS on a graph consisting of a circle with N half-lines attached at a single vertex, using the self-adjoint Laplacian with δ'-type vertex conditions (derivative continuity without function continuity). Solutions are built from dnoidal elliptic profiles on the circle matched to either zero tails or soliton tails on the half-lines, with full derivative continuity at the attachment point. Orbital stability is claimed for all nontrivial Z in the trivial-tail case; both existence and orbital (in)stability results are claimed in the nontrivial-tail case, depending on N, Z<0, and the phase velocity.

Significance. If the constructions and stability analysis are valid, the work provides explicit examples of standing waves on a hybrid graph with δ' interactions and extends orbital-stability techniques from simpler graphs to this setting. The explicit elliptic-function profiles and parameter-dependent stability thresholds would be useful for further studies of nonlinear waves on metric graphs.

major comments (2)
  1. [§3 (construction of standing waves)] The trivial-tail construction appears to violate the δ' vertex condition. With zero tails on the half-lines, both the function value and derivative vanish at the vertex, forcing the common derivative to be zero. The δ' condition then reduces to the sum of function values at the vertex equaling zero. The N zero tails contribute nothing, so the dnoidal profile on the circle must satisfy u(vertex)=0. However, derivative continuity with the zero tails requires the dnoidal profile to have vanishing derivative at the attachment point (an extremum), where the amplitude is nonzero. This yields a nonzero sum, contradicting the condition unless the amplitude is identically zero. This issue is load-bearing for the existence claim in the trivial-tail case (abstract and §3).
  2. [§4 (stability analysis)] The stability analysis for the trivial-tail case relies on the constructed profiles satisfying the domain D(H_Z). If the profiles do not lie in the domain, the orbital-stability statements (abstract) cannot hold. A concrete check of the vertex condition for the dnoidal profile at its maximum/minimum point is needed.
minor comments (2)
  1. [§2 (graph and operator)] Clarify the precise attachment geometry of the looping edge (whether the circle is attached at one point or two) and how the dnoidal function is defined across the full period.
  2. [§5] Add a brief remark on why the phase-velocity parameter appears only in the nontrivial-tail stability thresholds and not in the trivial-tail case.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and for identifying a critical inconsistency in the trivial-tail construction. We address the comments point by point below and will revise the manuscript to correct the affected claims.

read point-by-point responses

  1. Referee: [§3 (construction of standing waves)] The trivial-tail construction appears to violate the δ' vertex condition. With zero tails on the half-lines, both the function value and derivative vanish at the vertex, forcing the common derivative to be zero. The δ' condition then reduces to the sum of function values at the vertex equaling zero. The N zero tails contribute nothing, so the dnoidal profile on the circle must satisfy u(vertex)=0. However, derivative continuity with the zero tails requires the dnoidal profile to have vanishing derivative at the attachment point (an extremum), where the amplitude is nonzero. This yields a nonzero sum, contradicting the condition unless the amplitude is identically zero. This issue is load-bearing for the existence claim in the trivial-tail case (abstract and §3).

    Authors: We agree with the referee's analysis of the vertex conditions. For trivial tails, derivative continuity requires the dnoidal profile's derivative to vanish at the attachment point. With the common derivative equal to zero, the δ' condition reduces to the sum of function values equaling zero. Since the tails contribute zero, the dnoidal value at the vertex must be zero. However, a nontrivial dnoidal solution cannot simultaneously satisfy u=0 and u'=0 at an interior point without being identically zero. We therefore acknowledge that the claimed nontrivial trivial-tail standing waves do not exist. We will revise §3 to remove the existence and stability claims for this case, update the abstract, and restrict the results to the nontrivial-tail constructions (which require Z<0 and appear to satisfy the conditions). revision: yes

  2. Referee: [§4 (stability analysis)] The stability analysis for the trivial-tail case relies on the constructed profiles satisfying the domain D(H_Z). If the profiles do not lie in the domain, the orbital-stability statements (abstract) cannot hold. A concrete check of the vertex condition for the dnoidal profile at its maximum/minimum point is needed.

    Authors: We concur that orbital stability requires the profiles to lie in D(H_Z). Because the trivial-tail profiles fail to satisfy the δ' conditions (as detailed in the response to the first comment), the stability statements for this case are invalid. In the revised manuscript we will remove or qualify the stability analysis in §4 for trivial tails, provide explicit verification that the retained nontrivial-tail profiles belong to D(H_Z), and ensure all claims are supported by valid constructions. revision: yes

Circularity Check

0 steps flagged

No significant circularity; claims rest on explicit profile constructions and standard stability analysis

full rationale

The paper constructs standing waves by matching dnoidal profiles on the circle to zero or soliton tails on half-lines while enforcing derivative continuity and the δ' vertex condition encoded in D(H_Z). Orbital stability for the trivial-tail case and (in)stability results for the non-trivial case are asserted to follow from energy or spectral methods applied to these profiles. No quoted step reduces a prediction to a fitted input by construction, invokes a self-citation as the sole justification for a uniqueness theorem, or renames a known result as a new derivation. The central claims therefore remain independent of the inputs and are not forced by definition or self-reference.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The work rests on standard functional-analytic properties of the self-adjoint Laplacian with δ' conditions and on the known theory of Jacobian elliptic functions; no new free parameters or postulated entities are introduced.

axioms (2)
  • domain assumption The operator H_Z is self-adjoint on the domain encoding continuity of derivatives at the vertex without continuity of the function itself
    Defines the linear part of the NLS and is invoked to set up the standing-wave problem.
  • standard math Jacobian elliptic dnoidal functions satisfy the required ODE on the circle and can be matched to soliton or zero tails
    Used to construct explicit profiles that satisfy the nonlinear equation away from the vertex.

pith-pipeline@v0.9.0 · 5754 in / 1397 out tokens · 56348 ms · 2026-05-19T03:57:24.815124+00:00 · methodology

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Works this paper leans on

53 extracted references · 53 canonical work pages

  1. [1]

    Adami, C

    R. Adami, C. Cacciapuoti, D. Finco and D. Noja. Stable standing waves for a NLS on star graphs as local minimizers of the constrained energy, J. Differential Equations, 260, 7397–7415. 2016

    work page 2016

  2. [2]

    Adami, C

    R. Adami, C. Cacciapuoti, D. Finco and D. Noja. Variational properties and orbital stability of standing waves for NLS equation on a star graph, J. Differential Equations, 257, 3738–3777. 2014

    work page 2014

  3. [3]

    Adami, E

    R. Adami, E. Serra and P. Tilli. NLS ground states on graphs, Calc. Var. Partial Differential Equations, 54(1), 743–761. 2015

    work page 2015

  4. [4]

    Adami, E

    R. Adami, E. Serra and P. Tilli. Negative energy ground states for the L2-critical NLSE on metric graphs, Comm. Math. Phys., 352, 387–406. 2017

    work page 2017

  5. [5]

    Adami, E

    R. Adami, E. Serra and P. Tilli. Multiple positive bound states for the subcritical NLS equation on metric graphs, Calc. Var., 58(1), 5, 16pp. 2019. 34 J. ANGULO AND A. MU ˜NOZ

    work page 2019

  6. [6]

    J. Angulo. Stability theory for two-lobe states on the tadpole graph for the NLS equation, Nonlinearity, 37, 045015. 2024

    work page 2024

  7. [7]

    J. Angulo. Stability theory for the NLS on looping-edge graphs, Math. Z., 308, 19. 2024

    work page 2024

  8. [8]

    J. Angulo. Nonlinear stability of periodic traveling wave solutions to the Schr¨ odinger and the modified Korteweg-de Vries equations, JDE, 235, 1–30. 2007

    work page 2007

  9. [9]

    Angulo and M

    J. Angulo and M. Cavalcante. Nonlinear Dispersive Equations on Star Graphs, 32o Col´ oquio Brasileiro de Matem´ atica, IMPA. 2019

    work page 2019

  10. [10]

    Angulo and M

    J. Angulo and M. Cavalcante. Linear instability of stationary solitons for the Korteweg-de Vries equation on a star graph, Nonlinearity, 34, 3373–3410. 2021

    work page 2021

  11. [11]

    Angulo and M

    J. Angulo and M. Cavalcante. Dynamics of the Korteweg-de Vries equation on a balanced metric graph, Bull. Braz. Math. Soc. (N.S.). To appear. 2024

    work page 2024

  12. [12]

    Angulo and N

    J. Angulo and N. Goloshchapova. On the orbital instability of excited states for the NLS equation with the δ-interaction on a star graph, Discrete Contin. Dyn. Syst. A., 38(10), 5039–5066. 2018

    work page 2018

  13. [13]

    Angulo and N

    J. Angulo and N. Goloshchapova. Extension theory approach in the stability of the standing waves for the NLS equation with point interactions on a star graph, Adv. Differential Equations, 23(11–12), 793–846. 2018

    work page 2018

  14. [14]

    Angulo and R

    J. Angulo and R. Plaza. Instability of static solutions of the sine-Gordon equation on a Y -junction graph with δ-interaction, J. Nonlinear Sci., 31, 50. 2021

    work page 2021

  15. [15]

    Angulo and R

    J. Angulo and R. Plaza. Instability theory of kink and anti-kink profiles for the sine-Gordon on Josephson tricrystal boundaries, Physica D, 427, 133020. 2021

    work page 2021

  16. [16]

    Angulo and R

    J. Angulo and R. Plaza. Unstable kink and anti-kink profiles for the sine-Gordon on a Y -junction graph with δ′-interaction at the vertex, Math. Z., 300, 2885–2915. 2022

    work page 2022

  17. [17]

    A. H. Ardila. Orbital stability of standing waves for supercritical NLS with potential on graphs, Appl. Anal., 99(8), 1359–1372. 2020

    work page 2020

  18. [18]

    Berkolaiko and P

    G. Berkolaiko and P. Kuchment. Introduction to Quantum Graphs, Math. Surv. Monogr., 186, Amer. Math. Soc., Providence, RI. 2013

    work page 2013

  19. [19]

    Berkolaiko, J

    G. Berkolaiko, J. L. Marzuola and D. E. Pelinovsky. Edge-localized states on quantum graphs in the limit of large mass, Ann. Inst. H. Poincar ˜A© C Anal. Non Lin ˜A©aire, 38, 1295–1335. 2021

    work page 2021

  20. [20]

    Blank, P

    J. Blank, P. Exner and M. Havlicek. Hilbert Space Operators in Quantum Physics, 2nd ed., Theor. Math. Phys., Springer, New York. 2008

    work page 2008

  21. [21]

    Bourgain

    J. Bourgain. Global Solutions of Nonlinear Schr¨ odinger Equations,Amer. Math. Soc. Coll. Publ. 46. American Mathematical Society, Providence. 1999

    work page 1999

  22. [22]

    Burioni, D

    R. Burioni, D. Cassi, M. Rasetti, P. Sodano and A. Vezzani. Bose-Einstein condensation on inhomogeneous complex networks, J. Phys. B: At. Mol. Opt. Phys., 34, 4697–4710. 2001

    work page 2001

  23. [23]

    P. F. Byrd and M. D. Friedman. Handbook of Elliptic Integrals for Engineers and Scientists, Springer, Berlin, Heidelberg, 2. 1971

    work page 1971

  24. [24]

    Cacciapuoti, D

    C. Cacciapuoti, D. Finco and D. Noja. Topology induced bifurcations for the NLS on the tadpole graph, Phys. Rev. E, 91, 013206. 2015

    work page 2015

  25. [25]

    Cacciapuoti, D

    C. Cacciapuoti, D. Finco and D. Noja. Ground state and orbital stability for the NLS equation on a general starlike graph with potentials, Nonlinearity, 30(8), 3271–3303. 2017

    work page 2017

  26. [26]

    Cavalcante

    M. Cavalcante. The Korteweg-de Vries equation on a metric star graph, Z. Angew. Math. Phys., 69, Art. 124. 2018

    work page 2018

  27. [27]

    Cazenave

    T. Cazenave. Semilinear Schr¨ odinger Equations,Courant Lecture Notes in Mathematics, AMS. vol 10, Prov- idence, 2003

    work page 2003

  28. [28]

    G. P. Chuiko, O. V. Dvornik, S. I. Shyian and Y. A. Baganov. A new age-related model for blood stroke volume, Comput. Biol. Med., 79, 144–148. 2016

    work page 2016

  29. [29]

    Cr´ epeau and M

    E. Cr´ epeau and M. Sorine. A reduced model of pulsatile flow in an arterial compartment, Chaos Solitons Fractals, 34(2), 594–605. 2007

    work page 2007

  30. [30]

    K. J. Engel and R. Nagel. One parameter semigroups for linear evolution equations, Springer, New York. 2006

    work page 2006

  31. [31]

    P. Exner. Magnetoresonance on a lasso graph, Found. Phys., 27, Art. 171. 1997

    work page 1997

  32. [32]

    Exner and P

    P. Exner and P. ˇSeba. Free quantum motion on a branching graph, Rep. Math. Phys., 28, 7–26. 1989

    work page 1989

  33. [33]

    Exner and E

    P. Exner and E. ˇSereˇsov´ a.Appendix resonances on a simple graph, J. Phys. A, 27, 8269–8278. 1994

    work page 1994

  34. [34]

    F. Fidaleo. Harmonic analysis on inhomogeneous amenable networks and the Bose-Einstein condensation, J. Stat. Phys., 160, 715–759. 2015

    work page 2015

  35. [35]

    Grillakis, J

    M. Grillakis, J. Shatah and W. Strauss. Stability theory of solitary waves in the presence of symmetry, II, J. Funct. Anal., 94, 308–348. 1990

    work page 1990

  36. [36]

    Grillakis, J

    M. Grillakis, J. Shatah and W. Strauss. Stability theory of solitary waves in the presence of symmetry, I, J. Funct. Anal., 74(1), 160–197. 1987

    work page 1987

  37. [37]

    Kairzhan and D

    A. Kairzhan and D. E. Pelinovsky. Multi-pulse edge-localized states on quantum graphs, Anal. Math. Phys., 11, 171 (26pp). 2021

    work page 2021

  38. [38]

    Kairzhan, D

    A. Kairzhan, D. E. Pelinovsky and R. Goodman. Drift of spectrally stable shifted states on star graphs, SIAM J. Appl. Dyn. Syst., 18, 1723–1755. 2019

    work page 2019

  39. [39]

    Kairzhan, R

    A. Kairzhan, R. Marangell, D. E. Pelinovsky and K. Xiao. Existence of standing waves on a flower graph, J. Differential Equations, 271, 719–763. 2021. SELF-ADJOINT EXTENSIONS FOR SCHR ¨ODINGER-TYPE EQUATIONS 35

    work page 2021

  40. [40]

    Kairzhan, D

    A. Kairzhan, D. Noja and D. E. Pelinovsky. Standing waves on quantum graphs, J. Phys. A: Math. Theor., 55, 243001 (51pp). 2022

    work page 2022

  41. [41]

    Kuchment

    P. Kuchment. Quantum graphs, I. Some basic structures, Waves Random Media, 14, 107–128. 2004

    work page 2004

  42. [42]

    J. L. Marzuola and D. E. Pelinovsky. Ground state on the dumbbell graph, Appl. Math. Res. Express. AMRX, 1, 98–145. 2016

    work page 2016

  43. [43]

    D. Mugnolo. Mathematical Technology of Networks, Springer Proc. Math. Stat., 128, Bielefeld, December 2013. 2015

    work page 2013

  44. [44]

    Mugnolo, D

    D. Mugnolo, D. Noja and C. Seifter. Airy-type evolution equations on star graphs, Anal. PDE, 11, 1625–1652. 2018

    work page 2018

  45. [45]

    Mugnolo, D

    D. Mugnolo, D. Noja and C. Seifter. On solitary waves for the Korteweg-de Vries equation on metric star graphs, arXiv:2402.08368. 2024

    work page arXiv 2024

  46. [46]

    D. Noja. Nonlinear Schr¨ odinger equation on graphs: recent results and open problems, Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 372, 20130002 (20pp). 2014

    work page 2014

  47. [47]

    D. Noja, D. E. Pelinovsky and G. Shaikhova. Bifurcations and stability of standing waves in the nonlinear Schr¨ odinger equation on the tadpole graph,Nonlinearity, 28, 2343–2378. 2015

    work page 2015

  48. [48]

    Noja and D

    D. Noja and D. E. Pelinovsky. Standing waves of the quintic NLS equation on the tadpole graph, Calc. Var., 59, 173. 2020

    work page 2020

  49. [49]

    A. Pankov. Nonlinear Schr¨ odinger equations on periodic metric graphs,Discrete Contin. Dyn. Syst. A, 38(4), 697–714. 2018

    work page 2018

  50. [50]

    A. Pazy. Semigroups of linear operators and applications to partial differential equations, Vol. 44. Applied Mathematical Sciences, Springer-Verlag, New York, 1983

    work page 1983

  51. [51]

    Reed and B

    M. Reed and B. Simon. Methods of Modern Mathematical Physics. II. Fourier Analysis, Self-Adjointness, Academic Press, New York-London. xv+361 pp. 1975

    work page 1975

  52. [52]

    Schubert, C

    C. Schubert, C. Seifert, J. Voigt and M. Waurick. Boundary systems and (skew-) self-adjoint operators on infinite metric graphs, Math. Nachr., 288, 1776–1785. 2015

    work page 2015

  53. [53]

    Z. A. Sobirov, D. Babajanov and D. Matrasulov. Nonlinear standing waves on planar branched systems: Shrinking into metric graph, Nanosystems, 8, 29–37. 2017

    work page 2017