Linear Subspaces of Solutions Applied to Hirota Bilinear Equations

- Linear subspace of solution is applied to Boussinesq and Kadomtseve-Petviashvili (KP) equations using Hirota bilinear transformation. A sufficient and necessary condition for the existence of linear subspaces of exponential travelling wave solutions to Hirota bilinear equations is applied to show that multivariate polynomials whose zeros form a vector space can generate the desire Hirota bilinear equations with given linear subspaces of solutions and formulate such multivariate polynomials by using multivariate polynomials which have one and only one zero

A refined invariant subspace method and applications to evolution equations

The invariant subspace method is refined to present more unity and more diversity of exact solutions to evolution equations. The key idea is to take subspaces of solutions to linear ordinary differential equations as invariant subspaces that evolution equations admit. A two-component nonlinear system of dissipative equations was analyzed to shed light on the resulting theory, and two concrete examples are given to find invariant subspaces associated with 2nd-order and 3rd-order linear ordinary differential equations and their corresponding exact solutions with generalized separated variables.Comment: 16 page

Shortest hop multipath algorithm for wireless sensor networks

AbstractShortest hop or distance path is one of the most common methods used for relaying messages in a wide variety of networks. It provides an efficient message relaying to destination in terms of energy and time. There are many algorithms for constructing shortest hop or distance path. However, according to our knowledge, no algorithm for constructing a shortest hop multipath for wireless sensor networks (WSNs) has yet been proposed in the literature. In this paper, we propose a novel distributed shortest hop multipath algorithm for WSNs in order to generate energy efficient paths for data dissemination or routing. The proposed algorithm generates shortest hop braided multipath to be used for fault-tolerance or load-balancing. It guarantees the BFS tree and generates near optimal paths in O(V.D+V) message complexity and O(D2) time complexity regarding the communication costs towards the sink after termination of algorithm

Linear superposition and interaction of Wronskian solutions to an extended (2+1)-dimensional KdV equation

The main purpose of this work is to discuss an extended KdV equation, which can provide some physically significant integrable evolution equations to model the propagation of two-dimensional nonlinear solitary waves in various science fields. Based on the bilinear Bäcklund transformation, a Lax system is constructed, which guarantees the integrability of the introduced equation. The linear superposition principle is applied to homogeneous linear differential equation systems, which plays a key role in presenting linear superposition solutions composed of exponential functions. Moreover, some special linear superposition solutions are also derived by extending the involved parameters to the complex field. Finally, a set of sufficient conditions on Wronskian solutions is given associated with the bilinear Bäcklund transformation. The Wronskian identities of the bilinear KP hierarchy provide a direct and concise way for proving the Wronskian determinant solution. The resulting Wronskian structure generates $N$-soliton solutions and a few of special Wronskian interaction solutions, which enrich the solution structure of the introduced equation

Darboux transformations for linear operators on two dimensional regular lattices

Darboux transformations for linear operators on regular two dimensional lattices are reviewed. The six point scheme is considered as the master linear problem, whose various specifications, reductions, and their sublattice combinations lead to other linear operators together with the corresponding Darboux transformations. The second part of the review deals with multidimensional aspects of (basic reductions of) the four point scheme, as well as the three point scheme.Comment: 23 pages, 3 figures, presentation improve