A class of lump solutions of (2+1)-dimensional Boussinesq equation are obtained with the help of Maple by using Hirota bilinear method. Some contour plots with different determinant values are sequentially made to show that the corresponding lump solution tends to zero when the determinant approaches zero. The particular lump solutions with specific values of the involved parameters are plotted, as illustrative examples.
Soliton molecules have become one of the hot topics in recent years. In this article, we investigate soliton molecules and some novel hybrid solutions for the (2+1)-dimensional generalized Konopelchenko–Dubrovsky–Kaup–Kupershmidt (gKDKK) equation by using the velocity resonance, module resonance, and long wave limits methods. By selecting some specific parameters, we can obtain soliton molecules and asymmetric soliton molecules of the gKDKK equation. And the interactions among N-soliton molecules are elastic. Furthermore, some novel hybrid solutions of the gKDKK equation can be obtained, which are composed of lumps, breathers, soliton molecules and asymmetric soliton molecules. Finally, the images of soliton molecules and some novel hybrid solutions are given, and their dynamic behavior is analyzed.
In this paper, a number of exact solutions of ([Formula: see text])-dimensional combined AQ: Kindly check the short title. pKP-BKP equation are given. By taking into account the Hirota bilinear method and making use of quadratic function, a number of lump solutions are obtained. Combining quadratic function with exponential function, the interaction phenomenon between lump and soliton, and the lump-soliton solutions are obtained. When the trigonometric function is combined with the exponential function, the breather wave solutions are obtained. Finally, by picking appropriate parameter values, the dynamic properties of these exact solutions are presented by three-dimensional, density and contour plots.
Given a graph G with vertex set V (G) = V and edge set E(G) = E, let G l be the line graph and G c the complement of G. Let G 0 be the graph with V (G 0 ) = V and with no edges, G 1 the complete graph with the vertex set V , G + = G andand vertex v is incident (resp., not incident) to edge e in G. Given x, y, z ∈ {0, 1, +, −}, the xyz-transformation G xyz of G is the graph with the vertex set V (G xyz ) = V ∪ E and the edge set E(G xyz ) = E(G x ) ∪ E((G l ) y ) ∪ E(W ), where W = B(G) if z = +, W = B c (G) if z = −, W is the graph with V (W ) = V ∪ E and with no edges if z = 0, and W is the complete bipartite graph with parts V and E if z = 1. In this paper we obtain the Laplacian characteristic polynomials and some other Laplacian parameters of every xyz-transformation of an r-regular graph G in terms of |V |, r, and the Laplacian spectrum of G.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.