Poisson–Lie transformations and generalized supergravity equations

In this paper we investigate Poisson–Lie transformation of dilaton and vector field $${\mathcal {J}}$$ J appearing in generalized supergravity equations. While the formulas appearing in literature work well for isometric sigma models, we present examples for which generalized supergravity equations are not preserved. Therefore, we suggest modification of these formulas.

Group formalism of Lie transformations, exact solutions and conservation laws of nonlinear time-fractional Kramers equation

In this paper, we will concentrate on a systematic investigation of finding Lie point symmetries of the nonlinear [Formula: see text]-dimensional time-fractional Kramers equation via Riemann–Liouville and Caputo derivatives. By using the Lie group analysis method, the invariance properties and the symmetry reductions of the time-fractional Kramers equation are provided. It is shown that by using one of the symmetries of the underlying equation, it can be transformed into a nonlinear [Formula: see text]-dimensional fractional differential equation with a new dependent variable and the derivative in Erdélyi–Kober sense. Furthermore, we construct some exact solutions for the time-fractional Kramers equation using the invariant subspace method. In addition, adapting Ibragimov’s method, using Noether identity, Noether operators and formal Lagrangian, we construct conservation laws of this equation.

Symmetry’s Applications to Bihamonic Equation and the Generalized Solution of Elasticity Problem

The generalized elasticity solutions are obtained in this paper by symmetries from Lie transformations. The symmetries of the bihamonic equation are obtained by mathematica package SYM. The several invariant solutions are found by solving the corresponding ordinary differential equations from Lie transformations. The superposition of invariant solutions, corresponding to an eigenfunction expansion, could yield the generalized elasticity solutions in rectangular coordinates and polar coordinates, where the eigenvalues arises from the invariance of bihamonic equation.

CLASSICAL SOLUTIONS OF SIGMA MODELS IN CURVED BACKGROUNDS BY THE POISSON–LIE T-PLURALITY

Classical equations of motion for three-dimensional σ-models in curved background are solved by a transformation that follows from the Poisson–Lie T-plurality and transform them into the equations in the flat background. Transformations of coordinates that make the metric constant are found and used for solving the flat model. The Poisson–Lie transformation is explicitly performed by solving the PDE's for auxiliary functions and finding the relevant transformation of coordinates in the Drinfel'd double. String conditions for the solutions are preserved by the Poisson–Lie transformations. Therefore we are able to specify the type of σ-model solutions that solve also equations of motion of three-dimensional relativistic strings in the curved backgrounds. Some simple examples are given.