This paper describes the soliton surfaces approach to the Oriented Associativity equation for n=3 case. The equation of associativity arose from the 2D topological field theory. We constructed the surface associated with the Oriented Associativity equation for n=3 case equations using Sym-Tafel formula, which gives a connection between the classical geometry of manifolds immersed in R m and the theory of solitons. The so-called Sym-Tafel formula simplifies the explicit reconstruction of the surface from the knowledge of its fundamental forms, unifies various integrable nonlinearities and enables one to apply powerful methods of the soliton theory to geometrical problems. The soliton surfaces approach is very useful in construction of the so-called integrable geometries. Indeed, any class of soliton surfaces is integrable. Geometrical objects associated with soliton surfaces (tangent vectors, normal vectors, foliations by curves etc.) usually can be identified with solutions to some nonlinear models (spins, chiral models, strings, vortices etc.). We consider the geometry of surfaces immersed in Euclidean spaces. The Oriented Associativity equation plays a fundamental role in the theory of integrable systems. Such soliton surfaces for the Oriented Associativity equation for n=3 case are considered, and first and second fundamental forms of soliton surfaces are found for this case. Also, we study an area of surfaces for the Oriented Associativity equation for n=3 case.
This paper describes the hierarchy for N = 2 and n=3 case with an metric ƞ11≠0 when V0 = 0 of associativity equations. The equation of associativity arose from the 2D topological field theory. 2D topological field theory represent the matter sector of topological string theory. These theories covariant before coupling to gravity due to the presence of a nilpotent symmetry and are therefore often referred to as cohomological field theories. We give a description of nonlinear partial differential equations of associativity in 2D topological field theories as integrable nondiagonalizable weakly nonlinear homogeneous system of hydrodynamic type. The article discusses nonlinear equations of the third order for a function f = f(x,t)) of two independent variables x, t. In this work we consider the associativity equation for n=3 case with an a metric 0 11 . The solution of some cases of hierarchy when N = 2 and V0 = 0 equations of associativity illustrated
The Witten-Dijkgraaf-Verlinde-Verlinde (WDVV) equations, also called the associativity equations, is a system of nonlinear partial di�erential equations for one function, depending on a nite number of variables. The WDVV equations were introduced a few decades ago in the context of two-dimensional topological eld theories.The task of giving the associativity equations a geometric interpretation has two complementary aspects. On one side, can write these equations in a form that does not depend on the choice of the coordinates. On the other side, one must demand that the geometrical structure should be capable to select a class of a�nely related coordinates. The coordinate selection rule is important in the geometrization of the associativity equations. In this paper, we consider the soliton surface of the associativity equation. The equation of associativity originated from 2D topological field theory. 2D topological field theory represent the matter sector of topological string theory. These theories covariant before coupling to gravity due to the presence of a nilpotent symmetry and are therefore often referred to as cohomological field theories. The surface is constructed using Sym-Tafel formula, which is a connection between classical manifold geometry and soliton theory.The Sym-Tafel formula reconstructs a surface from knowledge of its fundamental forms, combines integrable nonlinearities, and allows the application of soliton theory methods to geometric problems. The soliton surfaces approach is necessary in the construction of so-called integrable geometries. Any class of soliton surfaces is integrable. Geometric objects associated with the surfaces of the solitons can usually be identified with the solutions to the strings. Thus in this work soliton surfaces for the associativity equation for n = 3 case with an metric η 11 ≠ 0 are considered, and first and second fundamental forms of soliton surfaces are found for this case. In addition, we study an area of surfaces for the associativity equation for n = 3 case with an metric η 11 ≠ 0.
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.
hi@scite.ai
10624 S. Eastern Ave., Ste. A-614
Henderson, NV 89052, USA
Copyright © 2024 scite LLC. All rights reserved.
Made with 💙 for researchers
Part of the Research Solutions Family.