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Let (M , ω) be a compact connected Kähler manifold of complex dimension four and let [ χ ] ∈ H 1 , 1 (M ; R). We confirm the conjecture by Collins–Jacob–Yau [8] of the solvability of the deformed Hermitian–Yang–Mills equation, which is given by the following nonlinear elliptic equation ∑ i arctan ⁡ (λ i) = θ ˆ , where λ i are the eigenvalues of χ with respect to ω and θ ˆ is a topological constant. This conjecture was stated in [8] , wherein they proved that the existence of a supercritical C -subsolution or the existence of a C -subsolution when θ ˆ ∈ [ ((n − 2) + 2 / n) π / 2 , n π / 2) will give the solvability of the deformed Hermitian–Yang–Mills equation. Collins–Jacob–Yau conjectured that their existence theorem can be improved to θ ˆ > (n − 2) π / 2 , where n is the complex dimension of the manifold. In this paper, we confirm their conjecture that when the complex dimension equals four and θ ˆ is close to the supercritical phase π from the right, then the existence of a C -subsolution implies the solvability of the deformed Hermitian–Yang–Mills equation. [ABSTRACT FROM AUTHOR]

In this paper, we prove that any C 1 -regular Hamiltonian stationary Lagrangian submanifold in a symplectic manifold is smooth. More broadly, we develop a regularity theory for a class of fourth order nonlinear elliptic equations with two distributional derivatives. Our fourth order regularity theory originates in the geometrically motivated variational problem for the volume functional, but should have applications beyond. [ABSTRACT FROM AUTHOR]

This paper studies the linear stability problem for solitary wave solutions of Hamiltonian PDEs. The linear stability problem is formulated in terms of the Evans function, a complex analytic function denoted by D (λ) , where λ is the spectral parameter. The main result is the introduction of a new factor, denoted Π, in the Pego & Weinstein (1992) derivative formula D ″ (0) = χ Π d I d c , where I is the momentum of the solitary wave and c is the speed. Moreover this factor turns out to be related to transversality of the solitary wave, modeled as a homoclinic orbit: the homoclinic orbit is transversely constructed if and only if Π ≠ 0. The sign of Π is a symplectic invariant, an intrinsic property of the solitary wave, and is a key new factor affecting the linear stability. The factor χ was already introduced by Bridges & Derks (1999) and is based on the asymptotics of the solitary wave. A supporting result is the introduction of a new abstract class of Hamiltonian PDEs built on a nonlinear Dirac-type equation, which model a wide range of PDEs in applications. Examples where the theory applies, other than Dirac operators, are the coupled mode equation in fluid mechanics and optics, the massive Thirring model, and coupled nonlinear wave equations. A calculation of D ″ (0) for solitary wave solutions of the latter class is included to illustrate the theory. [ABSTRACT FROM AUTHOR]

In this paper, we are concerned with the deformed Pearcey determinant det ⁡ (I − γ K s , ρ Pe) , where 0 ≤ γ < 1 and K s , ρ Pe stands for the trace class operator acting on L 2 (− s , s) with the classical Pearcey kernel arising from random matrix theory. This determinant corresponds to the gap probability for the Pearcey process after thinning, which means each particle in the Pearcey process is removed independently with probability 1 − γ. We establish an integral representation of the deformed Pearcey determinant involving the Hamiltonian associated with a family of special solutions to a system of nonlinear differential equations. Together with some remarkable differential identities for the Hamiltonian, this allows us to obtain the large gap asymptotics, including the exact calculation of the constant term, which complements our previous work on the undeformed case (i.e., γ = 1). It comes out that the deformed Pearcey determinant exhibits a significantly different asymptotic behavior from the undeformed case, which suggests a transition will occur as the parameter γ varies. As an application of our results, we obtain the asymptotics for the expectation and variance of the counting function for the Pearcey process, and a central limit theorem as well. [ABSTRACT FROM AUTHOR]