A new variant of the (2 + 1)-dimensional [(2 + 1)d] Boussinesq equation was recently introduced by J. Y. Zhu, arXiv:1704.02779v2, 2017; see eq. (3). First, we derive in this paper the one-soliton solutions of both bright and dark types for the extended (2 + 1)d Boussinesq equation by using the traveling wave method. Second, N -soliton, breather, and rational solutions are obtained by using the Hirota bilinear method and the long wave limit. Nonsingular rational solutions of two types were obtained analytically, namely: (i) rogue-wave solutions having the form of W-shaped lines waves and (ii) lump-type solutions. Two generic types of semi-rational solutions were also put forward. The obtained semi-rational solutions are as follows: (iii) a hybrid of a first-order lump and a bright one-soliton solution and (iv) a hybrid of a first-order lump and a first-order breather.

Recently, an integrable system of coupled (2 + 1)-dimensional nonlinear Schrödinger (NLS) equations was introduced by Fokas (eq. (18) in Nonlinearity 29, 319324 (2016)). Following this pattern, two integrable equations [eqs.(2) and (3)] with specific parity-time symmetry are introduced here, under different reduction conditions. For eq. (2), two kinds of periodic solutions are obtained analytically by means of the Hirota's bilinear method. In the long-wave limit, the two periodic solutions go over into rogue waves (RWs) and semi-rational solutions, respectively. The RWs have a line shape, while the semi-rational states represent RWs built on top of the background of periodic line waves. Similarly, semi-rational solutions consisting of a line RW and line breather are derived. For eq. (3), three kinds of analytical solutions,viz., breathers, lumps and semi-rational solutions, representing lumps, periodic line waves and breathers are obtained, using the Hirota method. Their dynamics are analyzed and demonstrated by means of threedimensional plots. It is also worthy to note that eq. (2) can reduce to a (1 + 1)-dimensional "reverse-space" nonlocal NLS equation by means of a certain transformation, Lastly, main differences between solutions of eqs. (2) and (3) are summarized.

The (2+1)-dimensional [(2+1)d] Fokas system is a natural and simple extension of the nonlinear Schrödinger equation (see eq. (2) in A. S. Fokas, Inverse Probl. 10 (1994) L19-L22). In this letter, we introduce its PT -symmetric version, which is called the (2 + 1)d nonlocal Fokas system. The N-soliton solutions for this system are obtained by using the Hirota bilinear method whereas the semi-rational solutions are generated by taking the long-wave limit of a part of exponential functions in the general expression of the N-soliton solution. Three kinds of semi-rational solutions, namely (1) a hybrid of rogue waves and periodic line waves, (2) a hybrid of lump and breather solutions, and (3) a hybrid of lump, breather, and periodic line waves are put forward and their rather complicated dynamics is revealed.

In this paper, the scriptPT‐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 2N‐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.

An integrable extension of the Kadomtsev-Petviashvili (KP) and Davey-Stewartson (DS) equations is investigated in this paper. We will refer to this integrable extension as the (4 + 1)-dimensional Fokas equation. The determinant expressions of soliton, breather, rational, and semi-rational solutions of the (4 + 1)-dimensional Fokas equation are constructed based on the Hirota's bilinear method and the KP hierarchy reduction method. The complex dynamics of these new exact solutions are shown in both three-dimensional plots and two-dimensional contour plots. Interestingly, the patterns of obtained highorder lumps are similar to those of rogue waves in the (1 + 1)-dimensions by choosing different values of the free parameters of the model. Furthermore, three kinds of new semi-rational solutions are presented and the classification of lump fission and fusion processes is also discussed. Additionally, we give a new way to obtain rational and semi-rational solutions of (3 + 1)-dimensional KP equation by reducing the solutions of the (4 + 1)-dimensional Fokas equation. All these results show that the (4 + 1)-dimensional Fokas equation is a meaningful multidimensional extension of the KP and DS equations. The obtained results might be useful in diverse fields such as hydrodynamics, nonlinear optics and photonics, ion-acoustic waves in plasmas, matter waves in Bose-Einstein condensates, and sound waves in ferromagnetic media.