We study differential equations -- not necessarily monic -- for Jacobi modular forms. As an application, we derive a second-order differential equation for the elliptic genus of a $K3$ surface involving the logarithmic derivative of the modular invariant $j(\tau)$, and we explicitly find a third-order equationwith a leading coefficient of weight 12 for the elliptic genus of any four-dimensional Calabi--Yau variety. Furthermore, we propose a simple recursive algorithm for investigating second-order differential equations with leading coefficient $E_4(\tau)$ or $E_6(\tau)$.
In this paper, we investigate the Riemann-Hilbert problem and soliton solutions of the defocusing AB system with nonzero boundary conditions and spatial parity symmetry. The Riemann-Hilbert problem is constructed based on the analytic properties and asymptotic behaviors of the modified Jost eigenfunctions and scattering coefficients. Under the nonzero boundary conditions limx→±∞|A(x,t)|=A0>0 and limx→±∞B(x,t)=0, the defocusing AB system has pairs of simple discrete eigenvalues {zn,zn∗} on the circle |z|=A02. The scattering matrix acquires an additional symmetry relation due to the even-odd parity of the initial data, so the discrete spectrum comes in quartets {zn,−zn∗,zn∗,−zn}. We present explicit N-soliton solutions in the reflectionless case. As an application, we derive dark-bright and dark-anti-bright soliton solutions by choosing different parameters.
Let (W,0)⊂(Cn,0) be an isolated hypersurface singularity defined by a holomorphic germ g:(Cn,0)→(C,0). For i≥1, let Jaci(g) be the ideal generated by all i-th order partial derivatives of g, and for p≥1 set Jp(g):=∑i=1pJaci(g). For k≥0, we consider the generalized local algebra Mk,p(W):=On/(〈g〉+mkJp(g)) and its derivation Lie algebra Lk,p(W):=Der(Mk,p(W),Mk,p(W)). The dimension λk,p(W):=dimCLk,p(W) is a numerical analytic invariant of (W,0); hence the numbers λk,p(W) form a family of analytic invariants extending the classical Yau number, which is recovered when (k,p)=(0,1). Focusing on the case (k,p)=(1,2), we compute λ1,2(W) for the binomial and trinomial weighted homogeneous isolated singularities. We also formulate the sharp upper estimates conjecture for λk,p(W) and prove it for a large class of singularities.
