Open research questions in Advanced Mathematical Physics Problems

What the literature leaves open

• The paper only considers a limited range of examples. The performance of MscaleDNNs is not fully understood. Further theoretical and computational research is necessary.

  Deep learning for the semi-classical limit of the Schrödinger equation · 2026 · DOI

• Further theoretical and computational research is necessary to fully understand the performance of MscaleDNNs. The proposed approach can be extended to other types of equations and problems. The use of other types of neural networks and methods can be explored.

  Deep learning for the semi-classical limit of the Schrödinger equation · 2026 · DOI

• The lack of global dispersive decay in non-translation-invariant operators. The difficulty of extending Strichartz estimates to more general operators and geometries.

  Local Dispersive and Strichartz Estimates for the Schrödinger Equation Associated to the Ornstein–Uhlenbeck Operator · 2026 · DOI

• The lack of global dispersive decay in non-translation-invariant operators presents significant challenges. There is a need for a comprehensive Strichartz theory for the Ornstein-Uhlenbeck operator.

  Local Dispersive and Strichartz Estimates for the Schrödinger Equation Associated to the Ornstein–Uhlenbeck Operator · 2026 · DOI

• The singular potential can be locally unbounded, making it challenging to analyze the equation. The convergence orders of exponential integrators for the equation are not well established. The paper needs to establish a unified framework for presenting the proof of the main results.

  Error Estimates of an Exponential Wave Integrator for the Nonlinear Schrödinger Equation with Singular Potential · 2026 · DOI

• To improve the accuracy of numerical methods for solving the nonlinear Schrödinger equation. To extend the results of the paper to other numerical methods. To apply the results of the paper to other areas of physics and chemistry.

  Error Estimates of an Exponential Wave Integrator for the Nonlinear Schrödinger Equation with Singular Potential · 2026 · DOI

• The paper faces the challenge of proving asymptotic stability for a nonlinear partial differential equation. The equation has a conserved energy and mass, which must be taken into account in the proof. The paper must also address the issue of the solution remaining close to a soliton for all positive times.

  On stabilization at a soliton for generalized Korteweg–De Vries pure power equation for any power p ∈ (1, 5) · 2026 · DOI

• The paper assumes that the solution remains close to a soliton for all positive times. The proof of asymptotic stability is obtained using a combination of the virial inequality method and Kato smoothing, which may not be applicable to other equations.

  On stabilization at a soliton for generalized Korteweg–De Vries pure power equation for any power p ∈ (1, 5) · 2026 · DOI

• The study of the direct and inverse problems for the Boussinesq equation on the half-line has not been initiated before. The existence of global solutions for the Boussinesq equation on the half-line is not proven.

  The Boussinesq equation on the half-line · 2026 · DOI

• The technical challenge is the analysis of the semigroup associated with the equation. The domain challenge is the lack of understanding of the behavior of solutions in Morrey spaces.

  Evolution equation with fractional Schrödinger operators: monotonicity and exponential decay of solutions in Morrey spaces · 2026 · DOI

• The paper only considers the case of fractional Schrödinger operators. The methodology is limited to the analysis of the semigroup associated with the equation.

  Evolution equation with fractional Schrödinger operators: monotonicity and exponential decay of solutions in Morrey spaces · 2026 · DOI

• The lack of a probabilistic global theory for the nonlinear Schrödinger equation with a Moser-Trudinger nonlinearity on compact surfaces. The need for a framework of Yudovich's argument for the uniqueness of PDEs posed in H^1.

  NLS with exponential nonlinearity on compact surfaces · 2026 · DOI

• To analyze the equation in supercritical regimes. To provide a numerical simulation of the equation.

  NLS with exponential nonlinearity on compact surfaces · 2026 · DOI

• To this date, this is the only existence proof that allows for an infinite-rank projection, whereas the uniqueness and regularity remain open problems.

  Mori-Zwanzig formalism: An existence proof for weak solutions of the orthogonal dynamics equation · 2026

• In this manuscript, we determine the critical exponent for the three-dimensional semilinear wave equation with strong damping and thereby resolve an open problem posed in 2014.

  The critical exponent for the three-dimensional semilinear wave equation with strong damping · 2026

• Part (iv) of Theorem 1 establishes decay ‖(φ, φₜ, φₓ)(t)‖_{L²({|x|≥t+R})} → 0 for the tail of solutions, but the precise decay rate as a function of time and spatial localization radius R for the interacting multi-kink case is not characterized.

  Asymptotic stability of the sine-Gordon kinks under perturbations in weighted Sobolev norms · 2026 · DOI

• The proof strategy in Appendix B uses the wave packet testing method from Ifrim-Tataru [31] but adapts it specifically for sine-Gordon with the weighted Sobolev norm framework. The applicability of this method to other nonlinear dispersive equations (e.g., nonlinear Klein-Gordon or modified KdV) with comparable weighted norm structures is not discussed.

  Asymptotic stability of the sine-Gordon kinks under perturbations in weighted Sobolev norms · 2026 · DOI

• To extend the study of the model to higher values of π. To use the technique of regularity structures to study other models. To improve the understanding of the dynamical sine-Gordon model.

  Global well-posedness of the dynamical sine-Gordon model up to 6π · 2026 · DOI

• The global well-posedness of the dynamical sine-Gordon model has only been proven up to 4π. The study of the model using regularity structures has been limited.

  Global well-posedness of the dynamical sine-Gordon model up to 6π · 2026 · DOI

• To study the large-time behavior of solutions to nonlocal dispersal equations with other types of initial data. To provide a numerical analysis of the solution. To study the applications of the results to population dynamics.

  Large-time behavior in a nonlocal heat equation with absorption. The absorption dominated case with fast decaying initial data · 2026 · DOI

• The paper identifies a gap in the literature on the large-time behavior of solutions to nonlocal dispersal equations. The paper identifies a need to study the limit profile of solutions to nonlocal dispersal equations.

  Large-time behavior in a nonlocal heat equation with absorption. The absorption dominated case with fast decaying initial data · 2026 · DOI

• There is a lack of rigorous justification of the WT's predictions. The paper identifies a gap in the existing literature and provides a new proof of the kinetic approximation.

  Kinetic Approximation for Equations of Discrete Turbulence in the Subcritical Case · 2026 · DOI

• Future research could explore further applications of the paper's uncertainty principle. Future research could also investigate the use of Wigderson's framework in other areas.

  An abstract uncertainty principle with applications · 2026 · DOI

• The paper identifies a gap in the existing literature regarding the uncertainty principle in the L^p setting. The paper aims to address this gap by proving a more general uncertainty principle.

  An abstract uncertainty principle with applications · 2026 · DOI

• The paper identifies a gap in the understanding of the behavior of solutions to the Euler-Poincaré equations. Prior work has not presented global existence and blow-up results for the Euler-Poincaré equations.

  Global Existence and Blow-Up for the Euler-Poincaré Equations with a Class of Initial Data · 2026 · DOI

Most-cited papers in Advanced Mathematical Physics Problems

• Existence of positive bound state solution for the nonlinear Schrödinger–Bopp–Podolsky system · Electronic journal of qualitative theory of differential equations · 2021 · 15 citations
• Periodic solutions with long period for the Mackey–Glass equation · Electronic journal of qualitative theory of differential equations · 2020 · 8 citations
• Local existence and blow up of solutions to a logarithmic nonlinear wave equation with time-varying delay · Studia Universitatis Babes-Bolyai Matematica · 2023 · 6 citations
• Periodic solutions of relativistic Liénard-type equations · Electronic journal of qualitative theory of differential equations · 2020 · 5 citations
• A general review on the NLS equation with point-concentrated nonlinearity · Communications in Applied and Industrial Mathematics · 2023 · 5 citations
• Remotely $c$-almost periodic type functions in ${\mathbb{R}}^{n}$ · Archivum Mathematicum · 2022 · 4 citations
• Existence and stability of standing waves for the Choquard equation with partial confinement · Topological Methods in Nonlinear Analysis · 2020 · 4 citations
• Well-posedness for a fourth-order equation of Moore–Gibson–Thompsontype · Electronic journal of qualitative theory of differential equations · 2021 · 4 citations
• Analytical and computational results for the decay of solutions of a damped wave equation with variable-exponent nonlinearities · Topological Methods in Nonlinear Analysis · 2022 · 3 citations
• A blow up of solutions for a system of Klein-Gordon equations with variable exponent. Theoretical and Numerical Results · Roczniki Polskiego Towarzystwa Matematycznego. Seria 3, Matematyka Stosowana/Matematyka Stosowana/Mathematica Applicanda · 2023 · 3 citations

Most recent work

• Asymptotic stability of the sine-Gordon kinks under perturbations in weighted Sobolev norms · Communications in Partial Differential Equations · 2026
• Multiple solutions for a class of Schrödinger–Bopp–Podolsky system with sublinear nonlinearity · Electronic journal of qualitative theory of differential equations · 2026
• Decay of Solutions of Nonlinear Dirac Equations · Communications in Mathematical Physics · 2026
• The Traveling Wave Problem for the Shallow Water Equations: Well-posedness and the Limits of Vanishing Viscosity and Surface Tension · Communications in Mathematical Physics · 2026
• Asymptotic expansions for conformal scalar curvature equations near isolated singularities · Calculus of Variations and Partial Differential Equations · 2026
• Long Time Behavior of the Modified One-Dimensional Derivative Schrödinger Equation · Analysis in Theory and Applications · 2026
• Global well-posedness of the dynamical sine-Gordon model up to 6π · The Annals of Probability · 2026
• The Zero-Dispersion Limit for the Benjamin–Ono Equation on the Circle · Communications in Mathematical Physics · 2026
• Large-time behavior in a nonlocal heat equation with absorption. The absorption dominated case with fast decaying initial data · Journal of Differential Equations · 2026
• Decay estimates and Strichartz estimates of the Schrödinger operator with finite rank perturbations · Journal of Differential Equations · 2026

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