Lump solutions are one of important solutions to partial differential equations, both linear and nonlinear. This paper aims to show that a Hietarinta-type fourth-order nonlinear term can create lump solutions with second-order linear dispersive terms. The key is a Hirota bilinear form. Lump solutions are constructed via symbolic computations with Maple, and specific reductions of the resulting lump solutions are made. Two illustrative examples of the generalized Hietarinta-type nonlinear equations and their lumps are presented, together with three-dimensional plots and density plots of the lump solutions.
Let G be a connected hypergraph with even uniformity, which contains cut vertices. Then G is the coalescence of two nontrivial connected sub-hypergraphs (called branches) at a cut vertex. Let
We reconsider the continuity of the Lyapunov exponents for a class of smooth Schrödinger cocycles with a
We study a functional modelling the progressive lens design, which is a combination of Willmore functional and total Gauss curvature. First, we prove the existence for the minimizers of this class of functionals among the class of revolution surfaces rotated by the curves y = f(x) about the x-axis. Then, choosing such a minimiser as background surfaces to approximate the functional by a quadratic functional, we prove the existence and uniqueness of the solution to the Euler-Lagrange equation for the quadratic functionals. Our results not only provide a strictly mathematical proof for numerical methods, but also give a more reasonable and more extensive choice for the background surfaces.
For diffusion processes, we extend various two-sided exit identities to the situation when the process is only observed at arrival times of an independent Poisson process. The results are expressed in terms of solutions to the differential equations associated with the diffusions generators.
Using the weak convergence method introduced by A. Budhiraja, P. Dupuis, and A. Ganguly [Ann. Probab., 2016, 44: 1723{1775], we establish the moderate deviation principle for neutral functional stochastic differential equations driven by both Brownian motions and Poisson random measures.
In this paper, we establish some Brauer-type bounds for the spectral radius of Hadamard product of two nonnegative tensors based on Brauer-type inclusion set, which are shown to be sharper than the existing bounds established in the literature. The validity of the obtained results is theoretically and numerically tested.
We consider a class of analytic area-preserving mappings Cm-smoothly depending on a parameter. Without imposing on any non-degeneracy assumption, we prove a formal KAM theorem for the mappings, which implies many previous KAM-type results under some non-degeneracy conditions. Moreover, by this formal KAM theorem, we can also obtain some new interesting results under some weaker non-degeneracy conditions. Thus, the formal KAM theorem can be regarded as a general KAM theorem for areapreserving mappings.
For all generic
We give a Brualdi-type Z-eigenvalue inclusion set of tensors, and prove that it is tighter than the inclusion set given by G. Wang, G. L. Zhou, and L. Caccetta [Discrete Contin. Dyn. Syst. Ser. B, 2017, 22: 187–198] in a special case. We also give an inclusion set for lk,s-singular values of rectangular tensors.
A finite group G is said to be a Bn-group if any n-element subset A = {a1, a2,..., an} of G satisfies