Optical solitons and periodic solutions of the (2 + 1)-dimensional nonlinear Schrödinger's equation

In this Letter, we consider the $(2+1)$-dimensional nonlinear Schrödinger's equation. With the aid of the Jacobian elliptic equation, we derive the exact bright soliton, dark soliton, singular soliton and periodic solutions of this equation expressed in terms of trigonometric functions, hyperbolic functions and Jacobian elliptic functions, respectively. Finally, for certain parametric values, we plot three dimensional graphics of modulus, real and imaginary parts of some solutions, which can help one better understand their dynamical behavior via their graphics analysis.     

On the Duffin–Kemmer–Petiau equation with linear potential in the presence of a minimal length

We point out an erroneous handling in the literature regarding solutions of the $(1+1)$-dimensional Duffin–Kemmer–Petiau equation with linear potentials in the context of quantum mechanics with minimal length. Furthermore, using Brau's approach, we present a perturbative treatment of the effect of the minimal length on bound-state solutions when a Lorentz-scalar linear potential is applied.     

Characteristics of the solitary waves and rogue waves with interaction phenomena in a (2 + 1)-dimensional Breaking Soliton equation

Under inquisition in this paper is a $(2+1)$-dimensional Breaking Soliton equation, which can describe various nonlinear scenarios in fluid dynamics. Using the Bell polynomials, some proficient auxiliary functions are offered to apparently construct its bilinear form and corresponding soliton solutions which are different from the previous literatures. Moreover, a direct method is used to construct its rogue wave and solitary wave solutions using particular auxiliary function with the assist of bilinear formalism. Finally, the interactions between solitary waves and rogue waves are offered with a complete derivation. These results enhance the variety of the dynamics of higher dimensional nonlinear wave fields related to mathematical physics and engineering.     

Supersymmetric extensions of Schrödinger-invariance

The set of dynamic symmetries of the scalar free Schrödinger equation in d space dimensions gives a realization of the Schrödinger algebra that may be extended into a representation of the conformal algebra in $d+2$ dimensions, which yields the set of dynamic symmetries of the same equation where the mass is not viewed as a constant, but as an additional coordinate. An analogous construction also holds for the spin-$\frac{1}{2}$ Lévy-Leblond equation. An $N=2$ supersymmetric extension of these equations leads, respectively, to a ‘super-Schrödinger’ model and to the $(3|2)$-supersymmetric model. Their dynamic supersymmetries form the Lie superalgebras $\mathfrak{osp}(2|2)⋉\mathfrak{sh}(2|2)$ and $\mathfrak{osp}(2|4)$, respectively. The Schrödinger algebra and its supersymmetric counterparts are found to be the largest finite-dimensional Lie subalgebras of a family of infinite-dimensional Lie superalgebras that are systematically constructed in a Poisson algebra setting, including the Schrödinger–Neveu–Schwarz algebra ${\mathfrak{sns}}^{(N)}$ with N supercharges. Covariant two-point functions of quasiprimary superfields are calculated for several subalgebras of $\mathfrak{osp}(2|4)$. If one includes both $N=2$ supercharges and time-inversions, then the sum of the scaling dimensions is restricted to a finite set of possible values.     

On the geometry of null hypersurfaces in Minkowski space

The present work is divided into three parts. First we study the null hypersurfaces of the Minkowski space ${\mathbb{R}}_1^{n+2}$, classifying all rotation null hypersurfaces in ${\mathbb{R}}_1^{n+2}$. In the second part we start our analysis of the submanifold geometry of the null hypersurfaces. In the particular case of the $(n+1)$-dimensional light cone, we characterize its totally umbilical spacelike hypersurfaces, show the existence of non-totally umbilical ones and give a uniqueness result for the minimal spacelike rotation surfaces in the $3$-dimensional light cone. In the third and final part we consider an isolated umbilical point on a spacelike surface immersed in the 3-dimensional light cone of ${\mathbb{R}}_1^4$ and obtain the differential equation of the principal configuration associated to this point, showing that every classical generic Darbouxian principal configuration appears in this context.