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An Integrable Nonlinear Wave Model: Darboux Transform and Exact Solutions
Issue:
Volume 12, Issue 1, February 2023
Pages:
1-8
Received:
25 January 2023
Accepted:
13 February 2023
Published:
23 February 2023
Abstract: Soliton equations are infinite-dimensional integrable systems described by nonlinear partial differential equations. In the mathematical theory of soliton equations, the discovery of integrability of these equations has greatly promoted the understanding of their generality, and thus promoted their rapid development. A key feature of an integrable nonlinear evolution equation is the fact that it can be expressed as the compatibility condition of two linear spectral problems, i.e., a Lax pair, which plays a crucial roles in the Darboux transformation. A major difficulty, however, is the problem of associating nonlinear evolution equations with appropriate spectral problems. Therefore, it is interesting for us to search for the new spectral problem and corresponding nonlinear evolution equations. In this paper, a new integrable nonlinear wave model and its integrable nonlinear reduction are presented by using the introduced 2 × 2 matrix spectral problem. Based on the resulting gauge transforms between the 2 × 2 matrix Lax pairs, Darboux transforms are derived for the integrable nonlinear wave model and its integrable nonlinear reduction, from which an algebraic algorithm for solving this integrable nonlinear wave model and its integrable nonlinear reduction is given. As an application of the Darboux transform, explicit exact solutions of the integrable nonlinear reduction are obtained, including solitons, breathers, and rogue waves.
Abstract: Soliton equations are infinite-dimensional integrable systems described by nonlinear partial differential equations. In the mathematical theory of soliton equations, the discovery of integrability of these equations has greatly promoted the understanding of their generality, and thus promoted their rapid development. A key feature of an integrable ...
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The Metric Dimension of Subdivisions of Lilly Graph, Tadpole Graph and Special Trees
Basma Mohamed,
Mohamed Amin
Issue:
Volume 12, Issue 1, February 2023
Pages:
9-14
Received:
28 October 2022
Accepted:
16 November 2022
Published:
16 March 2023
Abstract: Consider a robot that is navigating in a space represented by a graph and wants to know its current location. It can send a signal to find out how far it is from each set of fixed landmarks. We study the problem of computing the minimum number of landmarks required, and where they should be placed so that the robot can always determine its location. The set of nodes where the landmarks are located is called the metric basis of the graph, and the number of landmarks is called the metric dimension of the graph. On the other hand, the metric dimension of a graph G is the smallest size of a set B of vertices that can distinguish each vertex pair of G by the shortest-path distance to some vertex in B. The finding of the metric dimension of an arbitrary graph is an NP-complete problem. Also, the metric dimension has several applications in different areas, such as geographical routing protocols, network discovery and verification, pattern recognition, image processing, and combinatorial optimization. In this paper, we study the metric dimension of subdivisions of several graphs, including the Lilly graph, the Tadpole graph, and the special trees star tree, bistar tree, and coconut tree.
Abstract: Consider a robot that is navigating in a space represented by a graph and wants to know its current location. It can send a signal to find out how far it is from each set of fixed landmarks. We study the problem of computing the minimum number of landmarks required, and where they should be placed so that the robot can always determine its location...
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Distance Version of Forgotten Topological Index of Some Graph Products
Aruvi Mathivanan,
Maria Joseph
Issue:
Volume 12, Issue 1, February 2023
Pages:
15-25
Received:
27 June 2022
Accepted:
18 July 2022
Published:
23 March 2023
Abstract: In this paper, we have been found the exact values of distance version of Forgotten Topological index of various types of products such as strong, corona and tensor products of simple and connected graphs. Also we have been calculate the exact values of distance version of Forgotten Topological index of tensor product of path, cycle and complete graphs.
Abstract: In this paper, we have been found the exact values of distance version of Forgotten Topological index of various types of products such as strong, corona and tensor products of simple and connected graphs. Also we have been calculate the exact values of distance version of Forgotten Topological index of tensor product of path, cycle and complete gr...
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