Yulei Cao

We attempt to reduce the reverse space-time nonlocal Nonlinear Schrödinger (NLS) equation to a family of nonlocal Korteweg–de Vries (KdV) equations using the multiple-scale method. This approach offers an alternative pathway for constructing nonlocal equations besides the perspectives of integrability and coupled equation systems. In the present formulation, the constructed equations possess only P T -symmetric invariant solutions. Furthermore, these particular nonlocal KdV solitons can serve as approximate representations of the symmetric invariant solutions of the small-amplitude nonlocal NLS equation.

Crystalline SiGe combining the excellent properties of silicon and germanium plays a key role in modern electronics and optoelectronics technology. In contrast to the extensive researches on electronic and photoelectric properties, here we turn the attention on the mechanical property. By using the molecular dynamics simulations, the fracture behaviors of the [001], [110] and [111] orientated SiGe nanowire with and without twist grain boundary are investigated in the tension process. The results reveal that the twist grain boundary play an important role in the mechanical property of SiGe nanowire as it can inhibit the dislocation atoms from moving along the axial orientation. The fracture behavior of the SiGe nanowire with twist grain boundary is sensitive to the twist angle, radius and temperature. Both the Young’s modulus and maximum stress decrease as the twist angle increases from 8° to 19° or the temperature increases from 50 K to 600 K, while they are enhanced as the radius of nanowire increases. As for the fracture strain, it changes slightly with the radius of nanowire, but decreases with the increasing twist angle and temperature. Interestingly that, the fracture behavior occurs at the twist grain boundary regardless of the twist angle, radius and temperature. Our studies enrich the knowledge of mechanical characteristic of SiGe nanowire with twist grain boundary, which paves the way for processing and device design.

In this paper, a new general bilinear Bäcklund transformation and Lax pair for the (2+1)-dimensional shallow water wave equation are given in terms of the binary Bell polynomials. Based on this transformation along with introducing an arbitrary function, the multi-kink soliton, line breather, and multi-line rogue wave solutions on a non-flat constant background plane are derived. Further, we found that the dynamic pattern of line breather on the background of periodic line waves are similar to the two-periodic wave solutions obtained through a multi-dimensional Riemann theta function. Also, the generation mechanism and smooth conditions of the line rogue waves on the periodic line wave background are presented with long-wave limit method. Additionally, a family of new rational solutions, consisting of line rogue waves and line solitons, are derived, which have never been reported before. Furthermore, the present work can be directly applied to other nonlinear equations.

In this paper, the Maccari system is investigated, which is viewed as a two‐dimensional extension of nonlinear Schrödinger equation. We derive doubly localized two‐dimensional rogue waves on the dark solitons of the Maccari system with Kadomtsev–Petviashvili hierarchy reduction method. The two‐dimensional rogue waves include line segment rogue waves and rogue‐lump waves, which are localized in two‐dimensional space and time. These rogue waves are generated by the resonant collision of rational solitary waves and dark solitons, the whole process of transforming elastic collision into resonant collision is analytically studied. Furthermore, we also discuss the local characteristics and asymptotic properties of these rogue waves. Simultaneously, the generating conditions of the line segment rogue wave and rogue‐lump wave are also given, which provides the possibility to predict rogue wave. Finally, a new way to obtain the high‐order rogue waves of the nonlinear Schrödinger equation are given by proper reduction from the semi‐rational solutions of the Maccari system.

The celebrated Korteweg–de Vries and Kadomtsev–Petviashvili (KP) equations are prototypical examples of integrable evolution equations in one and two spatial dimensions, respectively. The question of constructing integrable evolution equations in three-spatial dimensions has been one of the most important open problems in the history of integrability. Here, we study an integrable extension of the KP equation in three-spatial dimensions, which can be derived using a specific reduction of the integrable generalization of the KP equation in four-spatial and two-temporal dimensions derived in (Phys. Rev. Lett. 96, (2006) 190201). For this new integrable extension of the KP equation, we construct smooth multi-solitons, high-order breathers, and high-order rational solutions, by using Hirota’s bilinear method.

In this paper, the ‐symmetric version of the Maccari system is introduced, which can be regarded as a two‐dimensional generalization of the defocusing nonlocal nonlinear Schrödinger equation. Various exact solutions of the nonlocal Maccari system are obtained by means of the Hirota bilinear method, long‐wave limit, and Kadomtsev–Petviashvili (KP) hierarchy method. Bilinear forms of the nonlocal Maccari system are derived for the first time. Simultaneously, a new nonlocal Davey‐Stewartson–type equation is derived. Solutions for breathers and breathers on top of periodic line waves are obtained through the bilinear form of the nonlocal Maccari system. Hyperbolic line rogue wave (RW) solutions and semirational ones, composed of hyperbolic line RW and periodic line waves are also derived in the long‐wave limit. The semirational solutions exhibit a unique dynamical behavior. Additionally, general line soliton solutions on constant background are generated by restricting different tau‐functions of the KP hierarchy, combined with the Hirota bilinear method. These solutions exhibit elastic collisions, some of which have never been reported before in nonlocal systems. Additionally, the semirational solutions, namely, (i) fusion of line solitons and lumps into line solitons and (ii) fission of line solitons into lumps and line solitons, are put forward in terms of the KP hierarchy. These novel semirational solutions reduce to ‐lump solutions of the nonlocal Maccari system with appropriate parameters. Finally, different characteristics of exact solutions for the nonlocal Maccari system are summarized. These new results enrich the structure of waves in nonlocal nonlinear systems, and help to understand new physical phenomena.