Develops SDYM equations on self-dual backgrounds using integration techniques.
problem Integrating SDYM equations on self-dual backgrounds.
method Combining SDYM integration schemes with multidimensional dispersionless integrable equations.
result Defines generating differential form and develops dressing scheme.
The L-equivalent counterpart of the M-LXIX equation is found. This L-equivalent equation is the Gauss-Codazzi equation which is integrable by the dressing method. This means that the M-LXIX equation is also integrable in this sense.
We prove that second-order hyperbolic Monge-Ampere equations for one function of two variables are connected to the wave equation by a Backlund transformation if and only if they are integrable by the method of Darboux at second order. One direction of proof, proving Darboux integrability, follows the implications of t…
Method finds differential equations for integrable billiard tables.
problem Finding differential equations for integrable billiard tables.
method Introducing a method to find differential equations for functions defining tables.
result Illustrated method in three billiard systems.
The paper explores conditions for solving the Killing equation in curved spaces and spacetimes.
problem Integrating the Killing equation in curved spaces and spacetimes.
method Prolongation of the Killing equation using Young symmetrizers to derive integrability conditions.
result Explicit integrability conditions for the Killing equation are provided, limiting the number of solutions.
This research improves deep neural networks for parameter identification and prediction in stochastic Volterra integral equations.
problem Parameter identification and prediction in Volterra integral equations driven by Gaussian noise.
method Improved deep neural networks framework that incorporates inter-output relationships into the loss function.
result The framework enhances parameter estimation accuracy and provides accurate solutions for modeling stochastic systems.
Integral geometry applied to steady Euler equations in 3D.
problem Understanding compactly supported solutions of steady Euler equations in 3D.
method Developed integral geometry for stationary Euler equations, deduced a differential equation for a function w on Grassmannian, and proved that w's annulation implies zero solutions. result Proved that locally supported solutions of the steady Euler equation in R3 are zero when w's annulation is true. Proves integrability of dispersionless Hirota type equations in 4D implies symplectic Monge-Ampere property.
problem Proving integrability of dispersionless Hirota type equations in 4D.
method Analyzing the symplectic Monge-Ampere property and deriving relations.
result Complete classification of integrable dispersionless PDEs of Hirota type in 4D.
Mathematical formulas for elliptic curve integrals solve anomaly equations.
problem Mathematical formulation of contact term singularities on elliptic curves.
method Residue formulas and holomorphic anomaly equations.
result Regularized integrals on elliptic curves satisfy holomorphic anomaly equations.
We give a review of the systematic construction of hierarchies of soliton flows and integrable elliptic equations associated to a complex semi-simple Lie algebra and finite order automorphisms. For example, the non-linear Schrödinger equation, the n-wave equation, and the sigma-model are soliton flows; and the equation…
Solves Monge-Ampère on non-integrable manifolds.
problem Existence and uniqueness of solutions to Monge-Ampère equation on non-integrable almost complex manifolds.
method Existence and uniqueness proved using the Monge-Ampère equation.
result Existence and uniqueness of solutions proved.
Paper extends Poincaré's work to stochastic differential equations.
problem Existence of first integrals in stochastic differential equations.
method Introduce two definitions of local first integrals for SDEs.
result Stochastic version of Poincaré non-integrability theorem.
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…
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.
Classifies vector equations with higher symmetries.
problem Classifying integrable vector equations with higher symmetries.
method Complete classification of isotropic vector equations of geometric type with higher symmetries.
result New examples of integrable multi-component systems and auto-Backlund transformations found.
Paper solves a complex portfolio selection problem with time-inconsistent preferences.
problem Time-inconsistent preferences in portfolio selection.
method Unified framework with minimal assumptions, proving existence and uniqueness of solution.
result Existence and uniqueness of square-integrable solution for the integral equation.
Researchers use Mellin-Barnes integrals to study trinomial equations and their braids.
problem Analyzing the roots of trinomial algebraic equations.
method Global analytic continuation and Mellin-Barnes integral representations.
result Precise description of the Galois group of trinomial equations.
Integrates Lax pair equations for a specific Lie algebra.
problem Low-dimensional Lie algebras of infinitesimal character ildeβ0. method Shows complete integrability of Lax pair equations.
result Proves complete integrability for certain Lie algebras.
This work integrates differentiation and integration in Physics-Informed Neural Networks.
problem Solving integro-differential equations and computing integral transforms.
method Augmenting Physics-Informed Neural Networks with automatic integration.
result Solving complex integral transforms and integro-differential equations.
The abstract shows how constant mean curvature surfaces in hyperbolic space are linked to the Liouville equation.
problem The constant mean curvature one equation for surfaces in hyperbolic 3 space.
method Group theoretic constructions and equivalence of quotient representations.
result The Darboux integrability of the equations shows that they admit equivalent quotient representations.
Integrability of the (2+1)-dimensional Gauss-Codazzi-Mainardi equation is considered. It is shown that this equation is the particular cases of the Yang-Mills-Higgs-Bogomolny and self-dual Yang-Mills equations.
We investigate integrable second order equations of the form F(u_{xx}, u_{xy}, u_{yy}, u_{xt}, u_{yt}, u_{tt})=0. Familiar examples include the Boyer-Finley equation, the potential form of the dispersionless Kadomtsev-Petviashvili equation, the dispersionless Hirota equation, etc. The integrability is understood as the…
The Poisson equation on manifolds plays an fundamental role in many applications. Recently, we proposed a novel numerical method called the Point Integral method (PIM) to solve the Poisson equations on manifolds from point clouds. In this paper, we prove the convergence of the point integral method for solving the Pois…
The paper explores connections between Veronese webs and integrable equations, revealing new symmetries and structures.
problem Understanding the relationships between Veronese webs and integrable equations.
method Established correspondence between Veronese three-dimensional webs and hyper-CR structures, used dispersionless Lax pairs to deform integrable equations, computed contact symmetries and Backlund transformations.
result Found new integrable equations and structures related to Veronese webs, linking finite-dimensional systems to dispersionless integrable PDEs.
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.
New method to derive integrable systems from existing Lax systems.
problem Deriving new integrable systems from existing ones.
method Systematic method of deriving new integrable systems from a given one.
result Examples of new integrable systems derived, including the dispersionless Hirota equation, the general heavenly equation, and the web equations.
Paper proves nonpositive boundary integral for Liouville's equation, zero only for discs.
problem Properties of Liouville's equation and boundary integrals.
method Polyhomogeneous expansions and rigidity/gap theorems.
result Boundary integral is nonpositive and zero only for discs.
This article is a local analysis of integrable GL(2)-structures of degree 4. A GL(2)-structure of degree n corresponds to a distribution of rational normal cones over a manifold M of dimension (n+1). Integrability corresponds to the existence of many submanifolds that are spanned by lines in the cones. These GL(2)-stru…
Novel approach to dissipative prolongations of multipeakons in Camassa-Holm equation.
problem Dissipative prolongations of multipeakons after collision time.
method Bi-Hamiltonian formulation and first integrals approach.
result An n-peakon after collision becomes an (n-1)-peakon with preserved momentum.
The nonlinear equations describing all the nonsingular pencils of metrics of constant Riemannian curvature are derived and the integrability of these nonlinear equations by the method of inverse scattering problem is proved. It is proved that all the nonsingular pairs of compatible metrics of constant Riemannian curvat…
Derives equations for systems with external forces using variational methods.
problem Deriving equations for systems subjected to external forces.
method Variational derivation of Euler-Poincaré equations using Poisson groupoid geometry.
result Derives variational error for numerical integrators of forced systems.
New insights into hyperelliptic divisors via integrable Hirota equations.
problem Understanding the geometry of hyperelliptic divisors.
method Proving the vanishing of genus 3 theta constants with even characteristics.
result Integrable Hirota equations specify the structure of hyperelliptic divisors.
We use Vessiot theory and exterior calculus to solve partial differential equations(PDEs) of the type uyy = F(x, y,u,ux,uy,uxx,uxy) and associated evolution equations. These equations are represented by the Vessiot distribution of vector fields. We develop and apply an algorithm to find the largest integrable sub-distr…
We study integrable non-degenerate Monge-Ampere equations of Hirota type in 4D and demonstrate that their symmetry algebras have a distinguished graded structure, uniquely determining the equations. This is used to deform these heavenly type equations into new integrable PDE of the second order with large symmetry pseu…
We study some conformally invariant integral equations using the method of moving spheres.
Introduces a new 2C extension of the heavenly equation.
problem Solving the general heavenly equation and its extensions.
method Infinite hierarchy of nonlocal symmetries and recursion operator.
result Solutions correspond to 4D hyper-para-Hermitian metrics.
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.
Study proves existence and uniqueness for differential equations with non-Lipschitz coefficients.
problem Existence and uniqueness for differential equations with non-Lipschitz coefficients.
method Relying on robust Itô integration, prove existence and uniqueness results.
result Existence and uniqueness for one-dimensional differential equations with non-Lipschitz coefficients.
Study the Lax equation in infinite-dimensional Lie algebras and Lie groups.
problem Investigate the Lax equation in infinite-dimensional Lie algebras and Lie groups.
method Derived integral expansions and generalized Baker-Campbell-Hausdorff formula for Lie groups.
result Explicit representation of product integral in terms of exponential map.
Let u be a function of n independent variables x^1, ..., x^n, and U=(u_{ij}) the Hessian matrix of u. The symplectic Monge-Ampere equation is defined as a linear relation among all possible minors of U. Particular examples include the equation det U=1 governing improper affine spheres and the so-called heavenly equatio…
Paper solves stock loan pricing with finite maturity using integral equations.
problem Valuation of margin-call stock loans with finite maturities.
method Fourier Sine transform and Volterra integral equation approach.
result Integral representation of margin-call stock loan value.
The paper proves gradient estimates for a weighted p-Laplacian equation on Riemannian manifolds.
problem Gradient estimates for a weighted p-Laplacian equation on Riemannian manifolds.
method Assumes a Sobolev inequality and integral Ricci bounds, proving local gradient estimates and Liouville type results.
result Proves local gradient estimates and Liouville type results on manifolds with lower bounds of Ricci curvature.
Study complex structures with totally real sections, providing integrability equations.
problem Existence and integrability of complex structures with totally real sections.
method Explicit integrability equations derived from fiberwise Taylor expansions.
result Detailed fiberwise Taylor expansions and integrability equations in a geometric case.
Motion of curves and surfaces in R3 lead to nonlinear evolution equations which are often integrable. They are also intimately connected to the dynamics of spin chains in the continuum limit and integrable soliton systems through geometric and gauge symmetric connections/equivalence. Here we point out the fact that…
We find normal forms for parabolic Monge-Ampere equations. Of these, the most general one holds for any equation admitting a complete integral. Moreover, we explicitly give the determining equation for such integrals; restricted to the analytic case, this equation is shown to have solutions. The other normal forms exha…
In [17] A. Rapcsák obtained necessary and sufficient conditions for the projective Finsler metrizability in terms of a second order partial differential equations. In this paper we investigate the integrability of the Rapcsák system, consisting of the Rapcsák equations and the homogeneity condition, by using the Spence…
Study integrability of conformal geodesics on gravitational instantons.
problem Integrability of conformal geodesic flow on gravitational instantons.
method Analyzing conformal geodesic flow equations on SO(3)-invariant gravitational instantons, separating Hamilton-Jacobi equations, finding commuting first integrals, and using conformal Killing-Yano tensors. result First example of an integrable conformal geodesic flow on a non-symmetric four-manifold.
The method of contact integrable extensions is used to find new zero-curvature representation for Plebañski's second heavenly equation.