Open research questions in Advanced Mathematical Physics Problems

What the literature leaves open

• The paper suggests that future research could focus on extending the results to larger perturbations. The paper suggests that future research could explore the applications of the sine-Gordon equation in various physical contexts.

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

• Future research can focus on the applications of the results to other types of differential equations. The authors suggest studying the case where n < 6. Future research can aim to improve the understanding of the behavior of solutions near isolated singularities.

  Asymptotic expansions for conformal scalar curvature equations near isolated singularities · 2026 · DOI

• 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 aims to address this gap by proving a more general uncertainty principle.

  An abstract uncertainty principle with applications · 2026 · DOI

• Domain challenges include the need for a comprehensive understanding of the behavior of coupled Klein-Gordon systems under various conditions. The analysis of the energy functional and its properties poses significant mathematical challenges.

  Blow-up and decay for coupled Klein–Gordon equations with Kirchhoff term and dispersive term · 2026 · DOI

• Lack of understanding of the behavior of coupled Klein-Gordon systems under certain conditions. Limited knowledge of the global existence, decay, and blow-up of solutions for such systems. Need for a more comprehensive analysis of the energy functional and its properties.

  Blow-up and decay for coupled Klein–Gordon equations with Kirchhoff term and dispersive term · 2026 · DOI

• The degeneracy of the term η_3^2 at η_2 = 0, which requires a special treatment. The need to establish a restriction estimate for surfaces of finite type 3, which is a gap in current research. The application of the estimate to the analysis of the discrete nonlinear Schrödinger equations (DNLS), which requires a deep understanding of the equation and its properties.

  Restriction estimates for 2D surfaces of finite type 3 and applications to dispersive equations · 2026 · DOI

• The case of surfaces of finite type 3 remains a gap in current research on the Fourier restriction problem. The existing literature has not provided a restriction estimate for these surfaces. This gap is significant because surfaces of finite type 3 have applications to dispersive equations, particularly the DNLS.

  Restriction estimates for 2D surfaces of finite type 3 and applications to dispersive equations · 2026 · DOI

• There is a gap in the literature regarding the convergence of the intermediate long wave equation to other equations in certain limits. Prior work has not established a priori bounds and equicontinuity of orbits for solutions to the equation. The paper seeks to address this gap.

  A priori bounds and equicontinuity of orbits for the intermediate long wave equation · 2026 · DOI

• The current approach is tailored to onedimensional solitary waves of a specific form. The analysis is limited to the chosen parametrization in the instability construction. The study does not explore alternative families of special solutions or perturbations.

  Ill-posedness estimation of the derivative schrödinger equation with a fourth-order power nonlinear term in H s · 2026 · DOI

• Future research may explore alternative families of special solutions or perturbations. The study of higher-dimensional analogues of the equation may be a fruitful area of research. The analysis of other power nonlinearities in the equation may be of interest.

  Ill-posedness estimation of the derivative schrödinger equation with a fourth-order power nonlinear term in H s · 2026 · DOI

• The case α = 1 has been previously studied, but the case of arbitrary α > 0 has not. The paper identifies a gap in the literature for the study of higher-order nonlinear recurrences.

  Asymptotic Expansions for a Class of High-Order Nonlinear Difference Equations With General Decay Rate · 2026 · DOI

• The paper suggests that future research could focus on the study of the Benjamin–Bona–Mahony–Burgers equation with large initial data. The paper also suggests that future research could focus on the study of similar equations.

  On periodic solutions of the Benjamin–Bona–Mahony–Burgers equation · 2026 · DOI

• The framework addresses the need for a unified, higher-dimensional setting for generalized Laplace-Stieltjes transforms. The approach fills a gap in the analysis of Cauchy problems and PDEs with distributional data in various engineering applications.

  Unifying Transforms for Singular Cauchy Problems in Quantum Logistics, Supply Chain Resilience, and Energy Systems · 2026 · DOI

• Future research can build upon the results of this paper to establish further regularity estimates and non-degeneracy of solutions. The porosity of the free boundary can be further analyzed.

  Improved Regularity Estimates for Degenerate or Singular Fully Nonlinear Dead-Core Systems and Hénon-Type Equations · 2026 · DOI

• The paper notes that prior work has focused on local asymptotic stability results.

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

• The paper does not provide a comprehensive study of the applications of the results. The authors assume certain flatness conditions on K. The paper does not address the case where n < 6.

  Asymptotic expansions for conformal scalar curvature equations near isolated singularities · 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 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

• 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

• The single-component NLS equation cannot accurately describe the interaction among multiple optical fields. The coupled NLS system does not fully incorporate higher-order effects.

  Long-Time Asymptotics for the n-Component Coupled Higher-Order Nonlinear Schrödinger Equation · 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 paper identifies a gap in the literature on the existence and stability of periodic solutions to the Benjamin–Bona–Mahony–Burgers equation. The paper aims to fill this gap by establishing the existence and uniqueness of the solution under certain conditions.

  On periodic solutions of the Benjamin–Bona–Mahony–Burgers equation · 2026 · DOI

Most-cited papers in Advanced Mathematical Physics Problems

• Nonspreading wave packets · American Journal of Physics · 1979 · 1,451 citations
• Decay functions · Social Networks · 2000 · 300 citations
• Multiple semiclassical standing waves for a class of nonlinear Schrödinger equations · Topological Methods in Nonlinear Analysis · 1997 · 138 citations
• Infinite Square-Well Potential with a Moving Wall · American Journal of Physics · 1969 · 112 citations
• A topological shooting method and the existence of kinks of the extended Fisher-Kolmogorov equation · Topological Methods in Nonlinear Analysis · 1995 · 64 citations
• Another look at Schrödinger equations with prescribed mass · Journal of Differential Equations · 2024 · 42 citations
• Existence and non existence of the ground state solution for the nonlinear Schroedinger equations with $V(\infty)=0$ · Topological Methods in Nonlinear Analysis · 2005 · 39 citations
• Normalized ground states for the Sobolev critical Schrödinger equation with at least mass critical growth · Nonlinearity · 2024 · 37 citations
• Global existence and blow-up results for an equation of Kirchhoff type on $\mathbb R^N$ · Topological Methods in Nonlinear Analysis · 2001 · 32 citations
• Exact Solutions of Classical Scalar Field Equations · Journal of Nonlinear Mathematical Physics · 2021 · 30 citations

Most recent work

• Long-Time Asymptotics of the Sawada–Kotera Equation on the Line · SIAM Journal on Mathematical Analysis · 2026
• The stability of forced waves for the Fisher-KPP equation in a degenerate shifting environment · Journal of Differential Equations · 2026
• On the well-posedness and stability of KdV-type solitary waves for the Zakharov-Ito system · Journal of Differential Equations · 2026
• Uniqueness of solutions for the logarithmic Schrödinger equation · Mathematische Annalen · 2026
• Orthonormal Strichartz inequalities and their applications on abstract measure spaces · Mathematische Zeitschrift · 2026
• The nonlinear Schrödinger equation with combined nonlinearities in 1D · Nonlinearity · 2026
• 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
• Local Dispersive and Strichartz Estimates for the Schrödinger Equation Associated to the Ornstein–Uhlenbeck Operator · Journal of Fourier Analysis and Applications · 2026
• Universality in the Small-Dispersion Limit of the Benjamin–Ono Equation · Communications in Mathematical Physics · 2026

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