In the context of integrable field theory with boundary, the integrable non-linear sigma models in two dimensions, for example, the O(N ), the principal chiral, the CP N −1 and the complex Grassmannian sigma models are discussed on a half plane. In contrast to the well known cases of sine-Gordon, non-linear Schrödinger and affine Toda field theories, these non-linear sigma models in two dimensions are not classically integrable if restricted on a half plane. It is shown that the infinite set of non-local charges characterising the integrability on the whole plane is not conserved for the free (Neumann) boundary condition. If we require that these non-local charges to be conserved, then the solutions become trivial.

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In this paper, we investigate the entanglement degree in the interaction of a two-level atom prepared in a momentum eigenstate with an additional Stark-shift. We use the quantum entropy as a measure of the entanglement degree. The constants of motion and the density matrix of the system are obtained. The effect of the detuning and the Stark-shift parameters when the atom is prepared initially in the upper state are investigated. It is found that the detuning and the Stark shift affected on the entanglement of the atomic system, and the influence of the center-of-mass motion play an important role in the entanglement degree.

This paper aims to formulate the fractional quasi-inverse scattering method.Also, we give a positive answer to the following question: can the Ablowitz-Kaup-Newell-Segur (AKNS) method be applied to the space-time fractional nonlinear differential equations? Besides, we derive the Bäcklund transformations for the fractional systems under study. Also, we construct the fractional quasi-conservation laws for the considered fractional equations from the defined fractional quasi AKNS-like system. The nonlinear fractional differential equations to be studied are the space-time fractional versions of the Kortweg-de Vries equation, modified Kortweg-de Vries equation, the sine-Gordon equation, the sinh-Gordon equation, the Liouville equation, the cosh-Gordon equation, the short pulse equation, and the nonlinear Schrödinger equation.

We calculate the energy density and energy distribution of Kantowski-Sachs space-time, using Einstein, Bergmann-Thomson and Landau-Lifshitz energymomentum complexes, in the theory of teleparallel gravity. A comparison of the results shows that the Einstein and Bergmann-Thomson definitions furnish a consistent result for the energy density and energy distribution, but the definition of Landau-Lifshitz does not concur with them. We show that the space-time under consideration gives a counterexample that the energy distribution is the same either in general relativity or teleparallel gravity.