Learning Latent Graph Geometry via Fixed-Point Schr\"odinger-Type Activation: A Theoretical Study

Dmitry Pasechnyuk-Vilensky; Martin Tak\'a\v{c}

arxiv: 2507.20088 · v3 · submitted 2025-07-27 · 💻 cs.LG · math-ph· math.MP· math.OC· stat.ML

Learning Latent Graph Geometry via Fixed-Point Schr\"odinger-Type Activation: A Theoretical Study

Dmitry Pasechnyuk-Vilensky , Martin Tak\'a\v{c} This is my paper

Pith reviewed 2026-05-19 03:06 UTC · model grok-4.3

classification 💻 cs.LG math-phmath.MPmath.OCstat.ML

keywords latent graph geometrySchrödinger-type dynamicsgraph-stationary networkssupra-graphsheaf-based architectureshypothesis classesstrong monotonicityimplicit layers

0 comments

The pith

Under finite-dimensional strong-monotonicity and admissible-lift assumptions, resolvent feed-forward networks, graph-stationary networks, supra-graph systems, and unitary sheaf architectures represent identical hypothesis classes.

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

The paper studies neural networks whose hidden layers arise as stationary states of dissipative Schrödinger-type dynamics defined on a learned latent graph. Each such layer is treated as a differentiable implicit graph operation, and the graph itself is optimized over a stratified moduli space equipped with a Kähler-Hessian metric that makes natural-gradient steps and stratum transitions well-posed. Multilayer constructions are shown to be equivalent to a single global stationary problem on a supra-graph, with a penalized relaxation whose stationary points recover the exact solution in the infinite-penalty limit. Reverse-mode differentiation likewise emerges as the adjoint of the exact global system. Under the stated monotonicity and lift assumptions the four listed architectures therefore share the same represented functions, so that complexity bounds can be expressed in terms of the sparse geometry of the latent graph or supra-graph rather than the ambient dense connectivity.

Core claim

Under finite-dimensional strong-monotonicity and admissible-lift assumptions, the corresponding represented hypothesis classes coincide among resolvent feed-forward networks, graph-stationary networks, supra-graph stationary systems, and sheaf-based architectures with unitary connection. The resulting structural identifications yield complexity bounds controlled by sparse graph or supra-graph geometry rather than dense ambient connectivity.

What carries the argument

Stationary state of a dissipative Schrödinger-type dynamics on a learned latent graph, optimized over the stratified moduli space of weighted graphs with a non-degenerate Kähler-Hessian metric.

If this is right

A multilayer stationary network is exactly equivalent to a global stationary problem on the corresponding supra-graph.
Penalized global relaxations converge to the exact stationary states as the penalty parameter tends to infinity.
Reverse-mode differentiation through the multilayer network is recovered as the adjoint of the exact global stationary system.
Complexity bounds for the shared hypothesis classes are governed by the sparsity pattern of the latent graph or supra-graph.

Where Pith is reading between the lines

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

The same stationary-dynamics construction might be applied to other families of implicit layers to obtain analogous unifications.
Once the latent graph is learned, downstream tasks could exploit the resulting sparse geometry for faster inference or reduced memory.
The supra-graph view suggests a systematic way to compose multiple graph layers while preserving the exact stationary equivalence.

Load-bearing premise

Finite-dimensional strong-monotonicity together with admissible-lift conditions hold for the latent graphs arising in typical tasks.

What would settle it

A concrete finite-dimensional example in which the hypothesis classes of the four architectures differ while the strong-monotonicity and admissible-lift conditions are satisfied, or a graph-learning task whose empirical complexity scales with ambient dimension rather than with the learned graph sparsity.

read the original abstract

We study neural architectures in which each hidden layer is defined by the stationary state of a dissipative Schr\"odinger-type dynamics on a learned latent graph. On stable branches, the local stationary problem defines a differentiable implicit graph layer. To learn the graph itself, we optimize over the stratified moduli space of weighted graphs and equip each stratum with a non-degenerate K\"ahler-Hessian metric that keeps natural-gradient descent and face crossing well posed. We then show that a multilayer stationary network is equivalent to an exact global stationary problem on a supra-graph, and that it admits a penalized global relaxation whose stationary states converge to the exact one as the penalty parameter tends to infinity. Reverse-mode differentiation is recovered as the adjoint of the exact global system, and the penalized adjoint converges to it in the same limit. Finally, under finite-dimensional strong-monotonicity and admissible-lift assumptions, the corresponding represented hypothesis classes coincide among resolvent feed-forward networks, graph-stationary networks, supra-graph stationary systems, and sheaf-based architectures with unitary connection. The resulting structural identifications yield complexity bounds controlled by sparse graph or supra-graph geometry rather than dense ambient connectivity.

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.

Desk Editor's Note private letter to a colleague

The paper unifies resolvent, graph-stationary, supra-graph, and sheaf networks via stationary Schrödinger dynamics on latent graphs, but the equivalences hinge on unquantified strong-monotonicity and lift assumptions.

read the letter

This paper's core move is to model hidden layers as stationary points of Schrödinger-type dynamics on a learned latent graph, then show that under specific assumptions the hypothesis classes line up with those of resolvent networks, supra-graph systems, and sheaf architectures. That alignment supposedly gives complexity bounds driven by the sparse graph structure rather than dense ambient connectivity.

Referee Report

1 major / 1 minor

Summary. The paper studies neural architectures in which each hidden layer is defined by the stationary state of a dissipative Schrödinger-type dynamics on a learned latent graph. It optimizes over the stratified moduli space of weighted graphs equipped with a non-degenerate Kähler-Hessian metric. The manuscript claims that a multilayer stationary network is equivalent to an exact global stationary problem on a supra-graph, that a penalized global relaxation converges to the exact stationary state as the penalty tends to infinity, that reverse-mode differentiation is recovered as the adjoint of the exact global system, and that under finite-dimensional strong-monotonicity and admissible-lift assumptions the represented hypothesis classes coincide among resolvent feed-forward networks, graph-stationary networks, supra-graph stationary systems, and sheaf-based architectures with unitary connection, yielding complexity bounds controlled by sparse graph or supra-graph geometry rather than dense ambient connectivity.

Significance. If the central equivalences and complexity bounds hold under the stated assumptions, the work offers a theoretical unification of several implicit graph-based architectures and a route to complexity control via latent graph sparsity. The geometric treatment of the graph moduli space and the adjoint analysis for implicit layers are potentially valuable contributions to the study of stationary neural networks.

major comments (1)

[Abstract] Abstract (final paragraph): The claim that the hypothesis classes coincide among the four architectures rests on finite-dimensional strong-monotonicity and admissible-lift assumptions, yet the manuscript provides neither a quantitative modulus of strong monotonicity nor a bound relating lift dimension to ambient dimension. Without such characterization it is impossible to assess how restrictive the assumptions are for standard activations or graph Laplacians, rendering the scope of the structural identification and the ensuing complexity bounds unclear.

minor comments (1)

[Abstract] The abstract introduces the term 'supra-graph' without a concise definition or reference to its construction from the multilayer stationary system; a brief inline definition would improve readability.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the thorough reading and for recognizing the potential value of the geometric and adjoint analyses. We address the single major comment below.

read point-by-point responses

Referee: [Abstract] Abstract (final paragraph): The claim that the hypothesis classes coincide among the four architectures rests on finite-dimensional strong-monotonicity and admissible-lift assumptions, yet the manuscript provides neither a quantitative modulus of strong monotonicity nor a bound relating lift dimension to ambient dimension. Without such characterization it is impossible to assess how restrictive the assumptions are for standard activations or graph Laplacians, rendering the scope of the structural identification and the ensuing complexity bounds unclear.

Authors: We agree that the manuscript states the finite-dimensional strong-monotonicity and admissible-lift assumptions only qualitatively. These conditions are the minimal hypotheses under which the stationary states exist, the implicit layers are differentiable, and the four hypothesis classes coincide exactly; the complexity bounds then follow directly from the geometry of the (supra-)graph. While explicit quantitative moduli and lift-dimension bounds are not derived in the current text, they can be obtained for concrete activations (e.g., scaled ReLU or tanh) and Laplacians by standard estimates on the minimal eigenvalue and Lipschitz constants. We will add a short subsection in the revised version that supplies such quantitative illustrations for representative activations and graph families, thereby clarifying the practical scope of the identifications. revision: yes

Circularity Check

0 steps flagged

No circularity: derivations build from stationary dynamics and explicit assumptions without reduction to inputs

full rationale

The paper constructs multilayer stationary networks from dissipative Schrödinger-type dynamics on latent graphs, establishes equivalence to global supra-graph stationary problems via penalized relaxation, recovers reverse-mode differentiation as the adjoint, and then invokes finite-dimensional strong-monotonicity plus admissible-lift assumptions to equate hypothesis classes across architectures. These steps are presented as forward derivations from the implicit layer definition and global relaxation limit; the final complexity bounds follow directly from the identified sparse geometry rather than presupposing the equivalence or fitting parameters to the target result. No self-definitional loops, fitted inputs renamed as predictions, or load-bearing self-citations appear in the derivation chain. The assumptions are stated explicitly in the concluding step and do not reduce the preceding constructions to tautologies.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 2 invented entities

The central claims rest on two domain-specific assumptions plus several newly introduced geometric and structural objects whose independent empirical support is not provided in the abstract.

axioms (2)

domain assumption Finite-dimensional strong-monotonicity assumption
Invoked to establish coincidence of hypothesis classes across architectures.
domain assumption Admissible-lift assumption
Required for the structural identifications between graph-stationary and sheaf-based models.

invented entities (2)

Supra-graph no independent evidence
purpose: Encodes the entire multilayer stationary network as one global stationary problem.
Introduced to prove equivalence between stacked local layers and a single exact global system.
Stratified moduli space of weighted graphs equipped with non-degenerate Kähler-Hessian metric no independent evidence
purpose: Provides a geometric setting in which natural-gradient descent and face-crossing remain well-posed while learning the latent graph.
New geometric structure postulated to make optimization over graph connectivity tractable.

pith-pipeline@v0.9.0 · 5751 in / 1718 out tokens · 42660 ms · 2026-05-19T03:06:20.539709+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

under finite-dimensional strong-monotonicity and admissible-lift assumptions, the corresponding represented hypothesis classes coincide among resolvent feed-forward networks, graph-stationary networks, supra-graph stationary systems, and sheaf-based architectures
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

nonlinear Schrödinger and Landau–Lifshitz dynamics with provable stable stationary solutions smoothly dependent on input data and graph weights

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

12 extracted references · 12 canonical work pages

[1]

Foundations of Computational Mathematics, 18(2): 399–431

Delaunay triangulation of manifolds. Foundations of Computational Mathematics, 18(2): 399–431. Bondy, J. A.; Murty, U. S. R.; et al. 1976.Graph theory with applications, volume

work page 1976
[2]

The Annals of Probability, 31(3): 1583–1614

Concen- tration inequalities using the entropy method. The Annals of Probability, 31(3): 1583–1614. Chung, F. R. 1997.Spectral graph theory, volume

work page 1997
[3]

Maz’ya, V

Prentice hall Upper Saddle River, NJ. Maz’ya, V . 2013.Sobolev spaces. Springer. Melcher, C.; and Ptashnyk, M

work page 2013
[4]

SIAM Journal on Mathematical Analysis , 45(1): 407–429

Landau-Lifshitz- Slonczewski equations: global weak and classical solutions. SIAM Journal on Mathematical Analysis , 45(1): 407–429. Copyright 2013 Elsevier B.V ., All rights reserved. Shalev-Shwartz, S.; and Ben-David, S. 2014.Understanding machine learning: From theory to algorithms . Cambridge university press. Shao, M.; Yang, Y .; and Zhao, L

work page 2013
[5]

If w = w0, G(ψ(+∞), w0) = 0

The equation on the stationary state is G(ψ, w) = −iH(w)ψ − γP ⊥ ψ D(ψ, w) = 0, where H(w) = ∆(w) + diag(|ψ0|2), D(ψ, w) = ∆(w)ψ + diag(|ψ|2 − |ψ0|2)ψ. If w = w0, G(ψ(+∞), w0) = 0 . G is smooth, since ∆(w) is linear w.r.t. w =⇒ L ∈ C ∞, projector P ⊥ ψ is analytical ifψ ̸= 0, the nonlinear term|ψ|2ψ is polynomial. Thus, G ∈ C ∞(H2(V ) × RE +, L2(V )). Con...

work page 2013
[6]

Consider the extended system d dtΦ(t, ψ0) = F (Φ(t, ψ0)), Φ(0, ψ0) = ψ0, where the right-hand side F ∈ C ∞

According to the definition of asymptotic stability (Khalil and Grizzle 2002), ∃δ > 0 and the Lyapunov function V (ψ) ≥ 0 such that: dV dt ≤ −β∥ψ − ψs∥2 H 1 , β > 0 for ∥ψ0 − ψ(+∞, ψ0)∥H 1 < δ. Consider the extended system d dtΦ(t, ψ0) = F (Φ(t, ψ0)), Φ(0, ψ0) = ψ0, where the right-hand side F ∈ C ∞. By the theorem on the smooth dependence of solutions on...

work page 2002
[7]

For large enough T , one has ∥Φ(T, ψ0) − ψ(+∞, ψ0)∥H 1 < ε

The mapping ψ0 7→ Φ(T, ψ0) is smooth. For large enough T , one has ∥Φ(T, ψ0) − ψ(+∞, ψ0)∥H 1 < ε. Since ψ(+∞, ψ0) is the uniform limit of smooth functions Φ(Tn, ψ0) for Tn → ∞, and the convergence is exponential, we have ψ(+∞, ψ0) ∈ C ∞. For ψ′ in the vicinity ψ(+∞): ψ(+∞, ψ0) = ψ(+∞) + Z ∞ 0 ∂Φ ∂t (t, ψ0)dt. Exponential convergence guarantees the converg...

work page 2013
[8]

Thus: ∂ψ(+∞) ∂w(e) H 2 ≤ C5 exp(−c · dG(i, j)) ∂H ∂w(e) ψ(+∞) H 2 ≤ C2 exp(−cρ)

This follows from the Combes-Thomas estimate for elliptic operators on manifolds (Combes and Thomas 1973). Thus: ∂ψ(+∞) ∂w(e) H 2 ≤ C5 exp(−c · dG(i, j)) ∂H ∂w(e) ψ(+∞) H 2 ≤ C2 exp(−cρ). Lemma 2 (Residual Bound)) . The residual r = k(ψ(+∞)) − y satisfies: E(X,y)∼D[|r|] ≥ γ1dG(i, j) − C6δ if e ∈ Etrue C7δ otherwise . Proof. For e ∈ Etrue: By finite propag...

work page 1973
[9]

By the nerve lemma (Edelsbrun- ner and Harer 2010):

for the δ-net V ⊂ G (δ < ρ/ 4). By the nerve lemma (Edelsbrun- ner and Harer 2010):

work page 2010
[10]

Step 1 dGH ((V, dG∗), G) ≤ C1δ: Since V is a δ-net in G and G∗ uses edges EG = {(u, v) : dG(u, v) < ρ/ 2}: - For any u, v ∈ V , dG∗(u, v) ≤ dG(u, v) + O(δ) (by triangle inequality)

Define the metric dG∗(u, v) = inf p:u→v P e∈p dG(u, v) (since w∗(e) = dG(u, v)−2 =⇒ 1 w∗(e) = dG(u, v)2, but we redefine d(e) = 1 w(e) here). Step 1 dGH ((V, dG∗), G) ≤ C1δ: Since V is a δ-net in G and G∗ uses edges EG = {(u, v) : dG(u, v) < ρ/ 2}: - For any u, v ∈ V , dG∗(u, v) ≤ dG(u, v) + O(δ) (by triangle inequality). - For any x ∈ G , ∃v ∈ V with dG(...

work page 2018
[11]

On the one hand, addition probability (for an edge e /∈ Et) of the true edge e ∈ Etrue is P true add (e) ≥ 1 − σ2 B∆2e , while for a spurious edge it is P spur add (e) ≤ σ2 B(∆′e)2

(Boucheron, Lugosi, and Massart 2003). On the one hand, addition probability (for an edge e /∈ Et) of the true edge e ∈ Etrue is P true add (e) ≥ 1 − σ2 B∆2e , while for a spurious edge it is P spur add (e) ≤ σ2 B(∆′e)2 . On the other hand, removal probability for the true edge, due to ℓ1-regularization, is P true remove(e) ≈ 0, since weights stabilize ab...

work page 2003
[12]

Real counterparts of NSE and LLE Diffusion System The complex NSE is replaced by a real reaction-diffusion system with similar potential and dissipation terms

Apply the standard Rademacher generalization bound (Bartlett and Mendelson 2002), tak- ing into account both the convergence event (with proba- bility ≥ 1 − ϵ) and the Rademacher bound (with probability ≥ 1 − δ). Real counterparts of NSE and LLE Diffusion System The complex NSE is replaced by a real reaction-diffusion system with similar potential and dis...

work page 2002

[1] [1]

Foundations of Computational Mathematics, 18(2): 399–431

Delaunay triangulation of manifolds. Foundations of Computational Mathematics, 18(2): 399–431. Bondy, J. A.; Murty, U. S. R.; et al. 1976.Graph theory with applications, volume

work page 1976

[2] [2]

The Annals of Probability, 31(3): 1583–1614

Concen- tration inequalities using the entropy method. The Annals of Probability, 31(3): 1583–1614. Chung, F. R. 1997.Spectral graph theory, volume

work page 1997

[3] [3]

Maz’ya, V

Prentice hall Upper Saddle River, NJ. Maz’ya, V . 2013.Sobolev spaces. Springer. Melcher, C.; and Ptashnyk, M

work page 2013

[4] [4]

SIAM Journal on Mathematical Analysis , 45(1): 407–429

Landau-Lifshitz- Slonczewski equations: global weak and classical solutions. SIAM Journal on Mathematical Analysis , 45(1): 407–429. Copyright 2013 Elsevier B.V ., All rights reserved. Shalev-Shwartz, S.; and Ben-David, S. 2014.Understanding machine learning: From theory to algorithms . Cambridge university press. Shao, M.; Yang, Y .; and Zhao, L

work page 2013

[5] [5]

If w = w0, G(ψ(+∞), w0) = 0

The equation on the stationary state is G(ψ, w) = −iH(w)ψ − γP ⊥ ψ D(ψ, w) = 0, where H(w) = ∆(w) + diag(|ψ0|2), D(ψ, w) = ∆(w)ψ + diag(|ψ|2 − |ψ0|2)ψ. If w = w0, G(ψ(+∞), w0) = 0 . G is smooth, since ∆(w) is linear w.r.t. w =⇒ L ∈ C ∞, projector P ⊥ ψ is analytical ifψ ̸= 0, the nonlinear term|ψ|2ψ is polynomial. Thus, G ∈ C ∞(H2(V ) × RE +, L2(V )). Con...

work page 2013

[6] [6]

Consider the extended system d dtΦ(t, ψ0) = F (Φ(t, ψ0)), Φ(0, ψ0) = ψ0, where the right-hand side F ∈ C ∞

According to the definition of asymptotic stability (Khalil and Grizzle 2002), ∃δ > 0 and the Lyapunov function V (ψ) ≥ 0 such that: dV dt ≤ −β∥ψ − ψs∥2 H 1 , β > 0 for ∥ψ0 − ψ(+∞, ψ0)∥H 1 < δ. Consider the extended system d dtΦ(t, ψ0) = F (Φ(t, ψ0)), Φ(0, ψ0) = ψ0, where the right-hand side F ∈ C ∞. By the theorem on the smooth dependence of solutions on...

work page 2002

[7] [7]

For large enough T , one has ∥Φ(T, ψ0) − ψ(+∞, ψ0)∥H 1 < ε

The mapping ψ0 7→ Φ(T, ψ0) is smooth. For large enough T , one has ∥Φ(T, ψ0) − ψ(+∞, ψ0)∥H 1 < ε. Since ψ(+∞, ψ0) is the uniform limit of smooth functions Φ(Tn, ψ0) for Tn → ∞, and the convergence is exponential, we have ψ(+∞, ψ0) ∈ C ∞. For ψ′ in the vicinity ψ(+∞): ψ(+∞, ψ0) = ψ(+∞) + Z ∞ 0 ∂Φ ∂t (t, ψ0)dt. Exponential convergence guarantees the converg...

work page 2013

[8] [8]

Thus: ∂ψ(+∞) ∂w(e) H 2 ≤ C5 exp(−c · dG(i, j)) ∂H ∂w(e) ψ(+∞) H 2 ≤ C2 exp(−cρ)

This follows from the Combes-Thomas estimate for elliptic operators on manifolds (Combes and Thomas 1973). Thus: ∂ψ(+∞) ∂w(e) H 2 ≤ C5 exp(−c · dG(i, j)) ∂H ∂w(e) ψ(+∞) H 2 ≤ C2 exp(−cρ). Lemma 2 (Residual Bound)) . The residual r = k(ψ(+∞)) − y satisfies: E(X,y)∼D[|r|] ≥ γ1dG(i, j) − C6δ if e ∈ Etrue C7δ otherwise . Proof. For e ∈ Etrue: By finite propag...

work page 1973

[9] [9]

By the nerve lemma (Edelsbrun- ner and Harer 2010):

for the δ-net V ⊂ G (δ < ρ/ 4). By the nerve lemma (Edelsbrun- ner and Harer 2010):

work page 2010

[10] [10]

Step 1 dGH ((V, dG∗), G) ≤ C1δ: Since V is a δ-net in G and G∗ uses edges EG = {(u, v) : dG(u, v) < ρ/ 2}: - For any u, v ∈ V , dG∗(u, v) ≤ dG(u, v) + O(δ) (by triangle inequality)

Define the metric dG∗(u, v) = inf p:u→v P e∈p dG(u, v) (since w∗(e) = dG(u, v)−2 =⇒ 1 w∗(e) = dG(u, v)2, but we redefine d(e) = 1 w(e) here). Step 1 dGH ((V, dG∗), G) ≤ C1δ: Since V is a δ-net in G and G∗ uses edges EG = {(u, v) : dG(u, v) < ρ/ 2}: - For any u, v ∈ V , dG∗(u, v) ≤ dG(u, v) + O(δ) (by triangle inequality). - For any x ∈ G , ∃v ∈ V with dG(...

work page 2018

[11] [11]

On the one hand, addition probability (for an edge e /∈ Et) of the true edge e ∈ Etrue is P true add (e) ≥ 1 − σ2 B∆2e , while for a spurious edge it is P spur add (e) ≤ σ2 B(∆′e)2

(Boucheron, Lugosi, and Massart 2003). On the one hand, addition probability (for an edge e /∈ Et) of the true edge e ∈ Etrue is P true add (e) ≥ 1 − σ2 B∆2e , while for a spurious edge it is P spur add (e) ≤ σ2 B(∆′e)2 . On the other hand, removal probability for the true edge, due to ℓ1-regularization, is P true remove(e) ≈ 0, since weights stabilize ab...

work page 2003

[12] [12]

Real counterparts of NSE and LLE Diffusion System The complex NSE is replaced by a real reaction-diffusion system with similar potential and dissipation terms

Apply the standard Rademacher generalization bound (Bartlett and Mendelson 2002), tak- ing into account both the convergence event (with proba- bility ≥ 1 − ϵ) and the Rademacher bound (with probability ≥ 1 − δ). Real counterparts of NSE and LLE Diffusion System The complex NSE is replaced by a real reaction-diffusion system with similar potential and dis...

work page 2002