The paper solves integrable systems of PDEs, including famous equations.
problem Constructing solutions for multicomponent integrable PDEs.
method Reduction to a finite-dimensional system, using Nijenhuis geometry.
result Animations of multi-component soliton and cnoidal solutions.
Gradient and Laplacian estimates for complex Monge-Ampère equations found.
problem Estimating solutions to complex Monge-Ampère equations with singularities.
method Integral method applied to obtain gradient and Laplacian estimates.
result Gradient and Laplacian estimates for the solution to the singular complex Monge-Ampère equation.
A solution of Hilberts fourth problem lead to integral equation of the type generalized cosine transform. The present paper considers the solution that integral equation by integral geometry methods and propose an inversion formula for reconstruction of Crofton measures from projective smooth Finsler metrics in R3.
The study shows how certain ODEs and integrals are regular under Borel summation.
problem Analyzing the regularity of solutions to ODEs and integration problems.
method Using geometric perspective on Laplace and Borel transforms, the study examines level 1 ODEs and exponential period integrals over Lefschetz thimbles.
result Solutions of certain ODEs and integration problems are Borel regular.
The paper examines partial regularity of Lipschitz solutions to minimal surface system.
problem Understanding the regularity of solutions to the minimal surface system.
method Investigation of stationary, integral weak, and viscosity solutions; interior gradient estimate using maximum principle.
result Partial regularity results for Lipschitz solutions, including interior gradient estimate.
The paper proves conditions for Darboux integrability in diagonal hydrodynamic systems.
problem Conditions for Darboux integrability in diagonal hydrodynamic systems.
method Proof of conditions using Laplace transformation sequences and geometric interpretations.
result Diagonal systems of hydrodynamic type are Darboux integrable if and only if the corresponding systems for commuting flows are Darboux integrable.
Paper constructs solutions for a class of overdetermined systems.
problem Constructing solutions for a class of overdetermined systems.
method Resolution of the solution sheaf, sufficient condition for global exactness, gluing techniques, local solvability of the Treves complex.
result Obtained a sufficient condition for global exactness, leading to gluing techniques for local solutions.
Study of Killing spinor-valued forms and their integrability conditions.
problem Understanding Killing spinor-valued forms and their properties.
method Detailed treatment of prolongation and integrability conditions, relating to curvature of the manifold.
result New solutions found that are not from tensor products of Killing spinors and Killing-Yano forms.
Paper solves quantum differential equations for projective bundles using Borel multitransforms.
problem Integration of quantum differential equations for P1-bundles. method Introduced Borel (α,β)-multitransforms to reconstruct solutions. result Quantum analog of Leray-Hirsch theorem for quantum cohomology of P1-bundles. The paper improves curvature estimates for Ricci flow solutions with bounded scalar curvature.
problem Proving curvature estimates for Ricci flow solutions with bounded scalar curvature.
method Localised weighted curvature integral estimates for solutions to Ricci flow.
result Integral curvature estimates imply a uniform bound on the spatial L2 norm of the Riemannian curvature tensor. We give new results concerning the Frobenius integrability and solution of evolution equations admitting travelling wave solutions. In particular, we give a powerful result which explains the extraordinary integrability of some of these equations. We also discuss "local" conservations laws for evolution equations in ge…
New Liouville-type results for CR Yamabe equation in Heisenberg group.
problem Characterizing solutions to CR Yamabe equation in Heisenberg group.
method Integral estimates combined with divergence formula.
result Liouville-type results for bounded solutions in n=2 and solutions with pointwise decay assumption in n≥3. Scroll structures on solutions of 4D integrable equations are involutive and governed by a dispersionless hierarchy.
problem Characterizing the geometry of solutions to 4D integrable equations.
method Defining rational normal scrolls and showing their involutivity.
result Involutive scroll structures are governed by a dispersionless integrable hierarchy.
We study conformally flat surfaces with prescribed Gaussian curvature, described by solutions u of the PDE: Δu(x)+K(x)exp(2u(x))=0, with K(x) the Gauss curvature function at $x\in\RR^2$. We assume that the integral curvature is finite. For radially symmetric K we introduce the notion of a least integrally curv…
The eigenvalue problem for the square integrable solutions is studied usually for elliptic equations. In this note we consider such a problem for the hyperbolic Klein-Gordon equation on Lorentzian manifolds. The investigation could help to answer the question why elementary particles have a discrete mass spectrum. An i…
Integrable flows on null curves in anti-de Sitter 3-space studied.
problem Analyzing integrable flows on null curves in anti-de Sitter 3-space.
method Formulated integrable flows related to the KdV hierarchy on null curves exploiting the geometry of anti-de Sitter 3-space.
result Explicitly found closed stationary solutions in terms of periodic solutions of a Lamé equation.
Jacobi solved geodesics on triaxial ellipsoids.
problem Finding the shortest path on a triaxial ellipsoid.
method Numerical evaluation of integrals and solving coupled equations.
result Solution for geodesics on triaxial ellipsoids.
We consider the problem of computing the integrable sub-distributions of the non-integrable Vessiot distribution of multi-dimensional second order partial differential equations (PDEs). We use Vessiot theory and solvable structures to find the largest integrable distributions contained in the Vessiot distribution assoc…
We study supersymmetric harmonic maps from the point of view of integrable system. It is well known that harmonic maps from R^2 into a symmetric space are solutions of a integrable system . We show here that the superharmonic maps from R^{2|2} into a symmetric space are solutions of a integrable system, more precisely …
New integral transforms solve multilayer heat equations.
problem Solving multilayer heat equations with moving boundaries.
method Expanding Dirac delta function in eigenfunctions, constructing oscillating integral transforms.
result Semi-analytical solutions for various problems.
By the classical Martingale Representation Theorem, replication of random vectors can be achieved via stochastic integrals or solutions of stochastic differential equations. We introduce a new approach to replication of random vectors via adapted differentiable processes generated by a controlled ordinary differential …
The paper solves the Integration Problem for principal connections.
problem Describing discrete connections associated with a principal connection.
method Using the Lie or derivative functor to induce connections on the principal bundle.
result For flat principal connections, the Integration Problem has a unique solution among flat discrete connections.
Elliptic systems are characterized by Darboux integrability.
problem Characterizing elliptic differential systems with holomorphic solutions.
method Using a complex manifold and associated holomorphic Pfaffian system.
result Elliptic systems are Darboux integrable under generic conditions.
The paper proves symmetry and classification of solutions to an integral equation in the Heisenberg group.
problem Symmetry and classification of solutions to a specific integral equation in the Heisenberg group.
method Moving plane method and Hardy-Littlewood-Sobolev inequality for the Heisenberg group.
result For subcritical p, no positive solutions exist; for critical p, solutions are cylindrical and unique. The paper studies magnetic geodesic flows on 2-surfaces with integrable structures.
problem Analyzing magnetic geodesic flows on 2-surfaces with additional integrals.
method Constructing exact solutions to semi-Hamiltonian systems of PDEs using generalized hodograph method and Legendre transformation.
result Exact solutions constructed for semi-Hamiltonian systems of PDEs.
Integrable hierarchies linked to F-manifolds with compatible connection.
problem Connecting integrable systems to geometric structures.
method Study F-manifolds with compatible connection and their relation to integrable hierarchies.
result F-manifolds with compatible connection classify n arbitrary functions of a single variable. The paper studies how to transform a sequence of cmc planes into a minimal surface.
problem Transforming a sequence of constant mean curvature planes into a minimal surface.
method Using algebraic-geometric correspondence and solving the Gauss-Codazzi equations.
result A sequence of solutions to the sinh-Gordon system converges to a solution of Liouville's equation, which is related to the Korteweg-de Vries system.
Problems on region choices for knot and link diagrams solved using Alexander numbering.
problem Existence of solutions for region choice problems on knot and link diagrams.
method Alexander numbering for regions, alternative proofs, necessary and sufficient conditions.
result Existence of solutions for region choice problems on link diagrams.
Classifies self-similar solutions for heat equations with positive speed.
problem Classifying self-similar solutions for semilinear heat equations.
method Analyzes the semilinear heat equation ut=Δu+∣u∣p−1u for p>1. result Finite time blowing up solutions converge to a positive constant after rescaling.
Paper develops methods for solving complex stochastic equations using Malliavin calculus.
problem Existence, uniqueness, and regularity of solutions to BSVIEs.
method Malliavin calculus for tackling diagonal processes and nonlinear dependence.
result Developed well-posedness results for BSVIEs, including probabilistic interpretation of PDEs and portfolio optimization.
We give time-slicing path integral formulas for solutions to the heat equation corresponding to a self-adjoint Laplace type operator acting on sections of a vector bundle over a compact Riemannian manifold with boundary. More specifically, we show that such a solution can be approximated by integrals over finite-dimens…
We introduce a general constructive setting of the density ratio estimation problem as a solution of a (multidimensional) integral equation. In this equation, not only its right hand side is known approximately, but also the integral operator is defined approximately. We show that this ill-posed problem has a rigorous …
A theorem proves integrability of Fréchet tangent distributions.
problem Integrability of Fréchet tangent distributions on manifolds.
method Introduced Condition W, applied variational approach, used differential forms.
result Existence and uniqueness of maximal foliations.
In this paper, we study the formal solution space of a nonlinear PDE in a fiber bundle. To this end, we start with foundational material and introduce the notion of a pfd structure to build up a new concept of profinite dimensional manifolds. We show that the infinite jet space of the fiber bundle is a profinite dimens…
The paper establishes Harnack inequalities for solutions of nonlinear parabolic equations on manifolds with integral Ricci curvature bounds.
problem Analyzing solutions of nonlinear parabolic equations on manifolds with specific curvature constraints.
method Establishing space-time gradient estimates and integrating them to find Harnack inequalities.
result Harnack inequalities for positive solutions of nonlinear parabolic equations under integral Ricci curvature bounds.
We consider solutions (M,g(t)), 0 <= t <T, to Ricci flow on compact, four dimensional manifolds without boundary. We prove integral curvature estimates which are valid for any such solution. In the case that the scalar curvature is bounded and T is finite, we show that these estimates imply that the (spatial) integral …
HyperCR Einstein--Weyl equations in 2+1 dimensions reduce to a pair of quasi-linear PDEs of hydrodynamic type. All solutions to this hydrodynamic system can be in principle constructed from a twistor correspondence, thus establishing the integrability. Simple examples of solutions including the hydrodynamic reductions …
Multipeakons are special solutions to the Camassa-Holm equation described by an integrable geodesic flow on a Riemannian manifold. We present a bi-Hamiltonian formulation of the system explicitly and write down formulae for the associated first integrals. Then we exploit the first integrals and present a novel approach…
In this note, we first prove that the solution of mean curvature flow on a finite time interval [0,T) can be extended over time T if the space-time integration of the norm of the second fundamental form is finite. Secondly, we prove that the solution of certain mean curvature flow on a finite time interval [0,T) …
Veronese webs are closely related to bi-Hamiltonian systems, as was shown by Gelfand and Zakharevich. Recently a correspondence between Veronese three-dimensional webs and three-dimensional Einstein-Weyl structures of hyper-CR type was established. The latter were parametrized by Dunajski and Krynski via the solutions …
Paper designs Poisson integrators using machine learning.
problem Designing integrators that preserve Poisson geometry.
method Reformulated as an optimization problem in Hamilton-Jacobi PDE, solved using machine learning.
result Machine learning approximates solutions to Hamilton-Jacobi PDE.
We show existence and uniqueness of solutions to the Monge-Ampere equation on compact almost complex manifolds with non-integrable almost complex structure.
Algorithm finds Liouvillian solutions for planar rational vector fields.
problem Finding Liouvillian solutions for planar rational vector fields.
method Algorithm to compute telescoper for specific foliations and rational vector fields.
result Algorithm finds Liouvillian solutions for planar rational vector fields, given a large enough complexity bound.
Develops methods to solve complex and real Hessian equations.
problem Solving complex and real Hessian equations on various domains.
method Introduces an ansatz to reduce PDEs to systems of ODEs, integrating via abelian integrals.
result Constructs entire solutions of arbitrary subcritical phase for dHYM/LYZ and special Lagrangian equations.
This work proposes an unsupervised neural network framework for solving combinatorial optimization problems on graphs.
problem Challenges in neural networks solving combinatorial optimization problems without labeled instances.
method Inspired by Erdos' probabilistic method, a neural network parametrizes a probability distribution over sets, optimizing it to find low-cost integral solutions.
result The method provides valid solutions to the maximum clique problem and local graph clustering, achieving competitive results.
The paper provides gradient estimates for solutions on manifolds with integral Ricci bounds.
problem Global regularity estimates for solutions of Δu=f on Riemannian manifolds. method Proves Lp-gradient estimates under integral Ricci bounds and constructs a counterexample. result Optimal constant lower bounds on Ricci curvature are shown in the pointwise sense.
Study solves HJB equations for time-inconsistent control problems.
problem Time-inconsistent deterministic linear quadratic control problems.
method Characterized solutions using Riccati equations with integral terms, proving uniqueness.
result Uniqueness of solutions to equilibrium HJB equations proved.
This paper proposes a new method to learn integration schemes for complex ODEs.
problem Learning efficient integration schemes for non-linear ODEs and their identification.
method A novel framework to learn integration schemes that minimize an integration-related cost function.
result The proposed learning-based approach provides integration schemes close to analytical solutions.