Date Published: February 12, 2018
Publisher: Public Library of Science
Author(s): Wei Liu, Jing Zhang, Xiliang Li, Xiao-Jun Yang.
In this paper, we investigate two types of nonlocal soliton equations with the parity-time (PT) symmetry, namely, a two dimensional nonlocal nonlinear Schrödinger (NLS) equation and a coupled nonlocal Klein-Gordon equation. Solitons and periodic line waves as exact solutions of these two nonlocal equations are derived by employing the Hirota’s bilinear method. Like the nonlocal NLS equation, these solutions may have singularities. However, by suitable constraints of parameters, nonsingular breather solutions are generated. Besides, by taking a long wave limit of these obtained soliton solutions, rogue wave solutions and semi-rational solutions are derived. For the two dimensional NLS equation, rogue wave solutions are line rogue waves, which arise from a constant background with a line profile and then disappear into the same background. The semi-rational solutions shows intriguing dynamical behaviours: line rogue wave and line breather arise from a constant background together and then disappear into the constant background again uniformly. For the coupled nonlocal Klein-Gordon equation, rogue waves are localized in both space and time, semi-rational solutions are composed of rogue waves, breathers and periodic line waves. These solutions are demonstrated analytically to exist for special classes of nonlocal equations relevant to optical waveguides.
Since Bender and Boettcher  showed that in the spectrum of the Hamiltonian, large amounts of non-Herimitan Hamiltons with Parity-time-symmetry (PT-symmetry) possess real and positive spectrum, the PT-symmetry has been an interesting topic in quantum mechanics and has significant impact. In general, a non-Hermitian Hamiltonian H = ∂xx + V(x) is called PT-symmetric if V(x) holds for V(x) = V*(−x). If set V(x, t) = p(x, t)p*(−x, t) in the Hamiltonian H above, then the Schrödinger equation ipt = Hp is PT-symmetric. In recent years, many works on PT-symmetry have been presented [2–6]. PT-symmetry has been widely applied to many areas of physics, such as optics [4, 7, 8], such as Bose-Einstein condensates , such as quantum chromodynamics , and so on.
The two dimensional nonlocal NLS equation is translated into the bilinear form
through the variable transformation
Here f, g are functions with respect to three variables x, y and t, and satisfy the condition
the asterisk denotes complex conjugation, and the operator D is the Hirota’s bilinear differential operator  defined by
where P is a polynomial of Dx, Dy, Dt, ⋯.
To using the Hirota bilinear method for constructing soliton solutions of the Eq (3), we consider a transformation different from that considered by Tajiri [78, 79]. Here we allow for nonzero asymptotic condition (u,v)→(2,β2+ϵ) as x, t → ∞, and look for solution in the form
where f, g are functions with respect to three variables x, y and t, and satisfy the condition
Obviously, u=2,v=β2+ϵ is a constant solution of the Eq (3), and under the transformation (20), the Eq (3) is cast into the following bilinear form
In this paper, we proposed two types of nonlocal soliton equations under PT symmetry conditions, namely, a two dimensional nonlocal NLS equation and a coupled nonlocal Klein-Gordon equation. By employing the Hiorta’s bilinear method, soliton and periodic line wave solutions were derived. Although these soliton solutions may have singularities, but smooth periodic line waves and breathers have been obtained by taking suitable choice of the parameters. For the two dimensional nonlocal NLS equation, line breathers are both periodic in x and y direction, see Fig 1. For the coupled nonlocal Klein-Gordon equation, breathers are localized in t direction and periodic in x direction, see Fig 4(b). In particular, a subclass of mixed solution consisting of breathers and periodic line waves is also generated see Fig 4(c). By taking a long wave limit of soliton solutions, the fundamental rogue wave solutions and semi-rational solutions have been generated. For the two dimensional nonlocal NLS equation, rogue wave solutions are line rogue waves, see Fig 2. The semi-rational solutions describe a line rogue wave and a line breather arising from the constant background together and then disappearing into the constant background again, see Fig 3. For the coupled nonlocal Klein-Gordon equation, except the rogue waves (see Fig 5(a)), semi-rational solutions describing the interactions between rogue waves, breathers and periodic line waves have also been generated. Three types of them are shown in Fig 5(b), 5(c) and 5(d). These nonlinear wave interactions lead to several interesting dynamics in physical systems, particularly, they are important in the formation of different wave structures. As there are few researches about the rogue waves of PT-symmetry systems, our research may help to promote the understanding of rogue wave phenomenon in PT-symmetry systems.