Yamabe solitons and gradient Yamabe solitons on three-dimensional N(k)-contact manifolds

If a three-dimensional [Formula: see text]-contact metric manifold [Formula: see text] admits a Yamabe soliton of type [Formula: see text], then the manifold has a constant scalar curvature and the flow vector field [Formula: see text] is Killing. Furthermore, either [Formula: see text] has a constant curvature [Formula: see text] or the flow vector field [Formula: see text] is a strict contact infinitesimal transformation. Also, we prove that if the metric of a three-dimensional [Formula: see text]-contact metric manifold [Formula: see text] admits a gradient Yamabe soliton, then either the manifold is flat or the scalar curvature is constant. Moreover, either the potential function is constant or the manifold is of constant sectional curvature [Formula: see text]. Finally, we have given an example to verify our result.

A Lie Symmetry Solutions of Sawada-Kotera Equation

In this article, the Lie Symmetry Analysis is applied in finding the symmetry solutions of the fifth order Sawada-Kotera equation. The technique is among the most powerful approaches currently used to achieveprecise solutions of the partial differential equations that are nonlinear. We systematically show the procedure to obtain the solution which is achieved by developing infinitesimal transformation, prolongations, infinitesimal generatorsand invariant transformations hence symmetry solutions of the fifth order Sawada-Kotera equation. Key Words- Lie symmetry analysis. Sawada-Kotera equation. Symmetry groups. Prolongations. Invariant solutions. Power series solutions. Symmetry solutions.

Blobbed topological recursion: properties and applications

We study the set of solutions (ωg,n)g⩾0,n⩾1 of abstract loop equations. We prove that ωg,n is determined by its purely holomorphic part: this results in a decomposition that we call “blobbed topological recursion”. This is a generalisation of the theory of the topological recursion, in which the initial data (ω0,1, ω0,2) is enriched by non-zero symmetric holomorphic forms in n variables (φg,n)2g−2+n>0. In particular, we establish for any solution of abstract loop equations: (1) a graphical representation of ωg,n in terms of φg,n; (2) a graphical representation of ωg,n in terms of intersection numbers on the moduli space of curves; (3) variational formulas under infinitesimal transformation of φg,n; (4) a definition for the free energies ωg,0 = Fg respecting the variational formulas. We discuss in detail the application to the multi-trace matrix model and enumeration of stuffed maps.

Lanczos Approach to Noether’s Theorem

If the action A=∫t1t2L(q,q,t)dt is invariant under the infinitesimal transformation t˜=t+ετ(q,t), q˜=qr+εζr(q,t), r-1,...,n with ε=constant≤1, then the Noether’s theorem permits to construct the corresponding conserved quantity. The Lanczos method accepts that ε=qn+1 is a new degree of freedom, thus the Euler-Lagrange equation for this new variable gives the Noether’s constant of motion.

Noether Symmetry and First Integral of Discrete Nonconservative and Nonholonimic Hamiltoinian System

The letter focuses on studying Noether symmetry and conserved quantity of discrete nonconservative and nonholonomic Hamiltonian system. Firstly, the discrete Hamiltonian canonical equations and discrete energy equations of nonconservative and nonholonomic Hamiltonian systems are derived with discrete Hamiltonian action. Secondly, based on the quasi-invariance of discrete Hamiltonian action and equation of lattice under the infinitesimal transformation with respect to time, generalized coordinates and generalized momentums, the discrete analogue of Noether’s identity and determining equation of lattice are obtained for the systems. Thirdly, the discrete analogues of Noether’s theorems and conserved quantities of the systems are presented. Finally, one example is discussed to illustrate the application of the results.