Abstract
Exact periodic and localized solutions of a nonlocal Mel′nikov equation are derived by the Hirota bilinear method. Many conventional nonlocal operators involve integration over a spatial or temporal domain. However, the present class of nonlocal equations depends on properties at selected far field points which result in a potential satisfying parity time symmetry. The present system of nonlocal partial differential equations consists of two dependent variables in two spatial dimensions and time, where the dependent variables physically represent a wave packet and an auxiliary scalar field. The periodic solutions may take the forms of breathers (pulsating modes) and line solitons. The localized solutions can include propagating lumps and rogue waves. These nonsingular solutions are obtained by appropriate choice of parameters in the Hirota expansion. Doubly periodic solutions are also computed with elliptic and theta functions. In sharp contrast with the local Mel′nikov equation, the auxiliary scalar field in the present set of solutions can attain complex values. Through a coordinate transformation, the governing equation can reduce to the Schrödinger–Boussinesq system.
Highlights
Nonlinear evolution equations (NLEEs) are widely applicable in a variety of intriguing phenomena in physical sciences and engineering, e.g., biophysics, condensed matter, fluids, optics, particle dynamics, and plasma physics [1,2,3,4,5,6,7,8]
We have introduced and investigated a nonlocal Mel′nikov equation with partial reverse space time, which constitutes a multidimensional version of the nonlocal Schrodinger–Boussinesq equation with a parity-time
By using the Hirota bilinear method, soliton solutions are obtained. These soliton solutions may have singularities, general n-breather solutions and mixed solutions consisting of breathers and periodic line waves can be derived under proper parameter constraints and lead to nonsingular solutions
Summary
Nonlinear evolution equations (NLEEs) are widely applicable in a variety of intriguing phenomena in physical sciences and engineering, e.g., biophysics, condensed matter, fluids, optics, particle dynamics, and plasma physics [1,2,3,4,5,6,7,8]. Aside from the global operator, the linear portion of the differential equation may be of the forms of the Duffing, heat, or inviscid Burgers equations We consider another class of nonlocal evolution equations where the intensity of the wave motion depends on values of the dependent variable at points in the far field. Inspired by the works of Ablowitz, Mussliman, and Fokas, we propose a partial reverse space-time nonlocal Mel′nikov equation: 3uyy − uxt − 3u2 + uxx + κφφ∗(− x, y, − t)xx 0, (4) iφy uφ + φxx, where u and φ are functions of x, y, and t. We derive families of rational and semirational solutions to the partial reverse spacetime nonlocal Mel′nikov equation (4) by using the Hirota bilinear method.
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