By virtue of Hirota’s bilinear method and Kadomtsev-Petviashvili hierarchy reduction technique, the general breather, soliton and rational solutions in the (2 + 1)-dimensional Hirota equation are constructed. These solutions are expressed in terms of Gram determinants and Schur polynomials. The Nth-order breather and soliton solutions contain 2N free complex parameters, while Nth-order rational ones possess N free complex parameters. By utilizing the Hermitian matrices, the range of free parameters is determined such that it ensures the regularity of these breather and soliton solutions. For the rational solutions, their non-singularity is proved and the parity-time-symmetric condition is derived. Furthermore, the rich dynamic patterns of breather, soliton and rational solutions are established by various choices of free parameters.
The ℤN-graded Toda lattices are introduced and investigated under both infinite and periodic boundary conditions. Initially, a hierarchy of integrable ℤN-graded Toda lattices is constructed using the technique of discrete zero curvature equations under infinite boundary conditions. The integrability of these lattices is demonstrated through their bi-Hamiltonian structures. Subsequently, particular emphasis is placed on the study of the ℤN-graded Toda lattice, the first nontrivial lattice in the hierarchy. It is discovered that this lattice can be represented in a Newtonian form with an exponential potential in the Flaschka-Manakov variables. Furthermore, the periodic ℤN-graded Toda lattice is identified as either a periodic Toda lattice or a set of independent periodic Toda lattices sharing the same periodicity. Finally, the complete integrability of the periodic ℤN-graded Toda lattice as a Hamiltonian system in the Liouville sense is established.
In this paper, the authors employ the -steepest descent method and Bäckhand transformation to investigate the asymptotic stability of Dirac solitons in the context of the massive Thirring model (MTM for short) system. They formulate the solution to the Cauchy problem for the MTM system in terms of the solution to a Riemann-Hilbert (RH for short) problem. This RH problem is decomposed into two components: A pure radiation solution and a soliton solution. As a direct outcome of this decomposition, they establish the asymptotic stability of Dirac solitons within the MTM system.
By using the implicit function, the authors prove the existence of solutions of the conformally covariant split system on compact three-dimensional Riemannian manifolds. They give rise to certain initial data for the Einstein-scalar system and the Einstein-Maxwell system.
In this paper, the author first defines a regular controlled Lagrangian (RCL for short) system on a symplectic fiber bundle, establishing a good expression of the dynamical vector field of an RCL system. This dynamical vector field synthesizes the Euler-Lagrange vector field and its changes under the actions of the external force and the control. Moreover, the author describes the RCL-equivalence, the RpCL-equivalence, and the RoCL-equivalence, proving regular point and regular orbit reduction theorems for the RCL system and the regular Lagrangian system with symmetry and a momentum map. Finally, as an application the author considers the regular point reducible RCL systems on a generalization of Lie group.
In this paper, the authors give a uniqueness theorem for the Dirichlet problem of minimal maps into general Riemannian manifolds with non-positive sectional curvature, improving [Lee, YI., Ooi, Y. S. and Tsui, MP., Uniqueness of minimal graph in general codimension, J. Geom. Anal., 29, 2019, 121–133, Theorem 5.2]. The proof of this theorem is based on the convexity of several functions in terms of squared singular values along the geodesic homotopy of two given minimal maps.
Let (Ωn+1, g) be an (n+1)-dimensional smooth compact connected Riemannian manifold with smooth boundary ∂Ω = Σ. Assume that the Ricci curvature of Ω is nonnegative and the principal curvatures of Σ are bounded from below by a positive constant c. In this paper, by constructing a new weight function, the authors obtain a lower bound of the first nonzero Steklov eigenvalue under the assumption that SecΩ ≥ −k, where k is a positive constant. The authors also extend this result to the Steklov-type eigenvalue problem of the weighted Laplacian on a metric measure space.
Let M be a Riemannian manifold. For p ∈ M, the tensor algebra of the negative part of the affinization of the tangent space TpM of M at p has a natural structure of a meromorphic open-string vertex algebra. These meromorphic open-string vertex algebras form a vector bundle over M with a connection. The author constructs a sheaf of meromorphic open-string vertex algebras on the sheaf of parallel sections of this vector bundle. Using covariant derivatives, he constructs a representation on the space of smooth functions of the algebra of parallel tensor fields. These representations are used to construct a sheaf of left -modules generated by the sheaf of smooth functions. In particular, the author obtains a meromorphic open-string vertex algebra VM as the global sections on M of the sheaf and a left VM-module WM as the global sections on M of the sheaf . He shows that the Laplacian on M is in fact a component of a vertex operator for the left VM-module WM restricted to the space of smooth functions.
In this paper, the authors introduce a Morse-theoretic condition under which the Levi form is allowed to have negative eigenvalues outside critical locus, and show that the existence of an exhaustion function satisfying such a condition leads to vanishing theorems.
The authors introduce the coupled discrete 2-component nonlinear Schrödinger equation with M-solutions and prove that this type of discrete equation is an integrable discretization of the integrable Manakov equation of mixed type. Moreover, the integrable discrete equation of 1-d Schrödinger flow to the pseudo-projective 2-space U(2, 1)/U(1, 1) × U(1) is shown to be a geometric realization of the integrable discrete Manakov equation of mixed type.
The authors revisit the eigenvalue problem of the Dirichlet Laplacian on bounded domains in complete Riemannian manifolds. By building on classical results like Li-Yau’s and Yang’s inequalities, they derive upper and lower bounds for eigenvalues. For the projective spaces and their minimal submanifolds, they also give explicit estimates on the lower bound for the eigenvalue of the Dirichlet Laplacian.
The authors show the stability of Hermitian Yang-Mills metrics under deformations of complex structures of either the Iwasawa or the Nakamura threefolds.
This paper considers overdetermined boundary problems. Firstly, the author gives a proof of the Payne-Schaefer conjecture about an overdetermined problem of sixth order in the two-dimensional case and under an additional condition for the case of dimension no less than three. Secondly, the author proves an integral identity for an overdetermined problem of fourth order which can be used to deduce Bennett’s symmetry theorem. Finally, the author proves a symmetry result for an overdetermined problem of second order by integral identities.
This paper considers maps from pseudo-Hermitian manifolds to Kähler manifolds and introduces partial energy functionals for these maps. First, the authors obtain a foliated Lichnerowicz type result on general pseudo-Hermitian manifolds, which generalizes a related result on Sasakian manifolds by Shen–Shen–Zhang (2013). Next, the authors investigate critical maps of the partial energy functionals, which are referred to as -harmonic maps and ∂b-harmonic maps. The authors give a foliated result for both - and ∂b-harmonic maps, generalizing a foliated result of Petit (2002) for harmonic maps. Then the authors are able to generalize Siu’s holomorphicity result for harmonic maps by Siu (1980) to the case for - and ∂b-harmonic maps.
In this paper, the authors investigate the geometric rigidity of Riemannian manifolds under suitable curvature restrictions. The authors first prove a new gap theorem for the Ricci curvature of compact locally conformally flat Riemannian manifolds. Subsequently, the authors consider the Riemannian manifolds with the Cotton tensor C satisfying div C = 0 and prove some integral curvature pinching theorems.