Research Topics:  

• Damped Nonlinear Schrödinger Equation 

• Nonlinear Schröinger Equation on Graphs and guided structures

• Normalized Ginzburg-Landau and Heat Equations

Short description:

I am currently a post-doc in applied mathematics at the University of Toulouse under the supervision of Stefan Le Coz, with the grant of the laboratory of excellence CIMI. 

Our project focuses on studying the nonlinear Schrödinger equation on quantum graphs and the dimension reduction from higher-dimensional structures. 

At the same time, I collaborate with David Lafointaine on considering dissipative variants of dispersive equations with trapping regions. 

I also collaborate with Hiroaki Kikuchi on analysing the bifurcation points for the dimension reductions.

I completed my PhD in Applied Mathematics at the Gran Sasso Science Institute, L'Aquila, one of the 7 special status universities in Italy, under the supervision of Professor Paolo Antonelli. 

My thesis explored the stability of solitary waves in nonlinear Schrödinger-type equations with dissipation.

I obtained a Master’s Degree in Mathematics from the University of Pisa, Italy, under the supervision of Vladimir Georgiev.

We established the well-posedness of the nonlinear Dirac equation with nonlocal interactions in 2 dimensions, later published

Preprints :

1. Durand-Simonnet, E. and Shakarov, B., Existence and stability of ground states for the defocusing nonlinear Schr¨odinger equation on Quantum Graphs, arXiv :2502.18014 (2025)

2. Le Coz, S. and Shakarov, B., Ground states on a fractured strip and one-dimensional reduction, arXiv :2411.18187 (2024)

Publications :

1. Lafontaine, D. and Shakarov, B., Scattering for defocusing cubic NLS under locally damped strong trapping, arXiv :2502.06306 (2025) accepted in Ann. Henri Lebesgue (2025) 

2. Antonelli, P. and Shakarov, B., On the formation of singularity for the slightly supercritical NLS equation with nonlinear damping, J. Dyn. Differ. Equ.  arXiv :2309.08281  (2025), link

3. Shakarov, B., Global Solutions and Asymptotic Behavior to a Norm-preserving Non-local Parabolic Flow, Boll. Unione Mat. Ital. (2025) link

4. Antonelli, P. and Shakarov, B., Stability of cnoidal waves for the damped nonlinear Schrödinger equation, J. Hyperbolic Differ. Equ. 21(4) : 1045–1086, 2024 link

5. Antonelli, P. and Shakarov, B., Existence and large time behavior for a dissipative variant of the rotational NLS equation, Commun. Math. Sci. 22(6) : 1601–1633, 2024 link

6. Antonelli, P., Cannarsa, P. and Shakarov, B., Existence and asymptotic behavior for L^2-norm preserving nonlinear heat equations, Calc. Var. 63, 108 (2024) link

7. Georgiev, V. and Shakarov, B., Global large data solutions for 2D Dirac equation with Hartree-type interaction, Int. Math. Res. Not., 2022(17) : 12803–12820, 2022 link

8. Georgiev, V. and Shakarov, B., The boson star equation with Hartree-type nonlinearity : global existence in H^{1/2}(R^2),  PLISKA, 2019 link

Current projects:

• Shrinking limit from Quantum Books to Quantum Graphs:
  We study the limit transition from open book structures to quantum graphs (in collaboration with S. Le Coz, IMT).

• Bifurcation Points on fractured strips:
  Rigorous identification of the bifurcation point in the case of strips and quantum books (with H. Kikuchi, Tokyo).

• Quantum Graphs with Trapping Core:
  Scattering on quantum graphs with localized dissipation and a trapping core (with D. Lafontaine, IMT). 

• Defocusing NLS on Quantum Graphs:
  Following our recent work on action ground states with general decoupling conditions at the vertices, we study the existence conditions for energy ground states as well as for nodal ground states (in collaboration with É. Durand-Simonnet (IMT) and D. Galant (Valenciennes)).