Families of Rational and Semirational Solutions of the Partial Reverse Space-Time Nonlocal Mel′nikov Equation

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.

Cauchy matrix structure of the Mel’nikov model of long–short wave interaction

“…Both models can be viewed as complex reductions of the KPSCS (1.2). We have found that each short-wave packet f j 2 | | is actually characterized by an arbitrary function β j (t), which was never reported in the literatures on the research of the Mel'nikov model (1.1) and its extensions [1][2][3][4][22][23][24][25][26]. This b t j ( ) governs the effect of the shortwave packet and can bring more variety of interactions of line solitons.…”

Field and reverse field solitons in wave-operator nonlinear Schrödinger equation with space-time reverse: Modulation instability

“…The nonlocal complex STR modified Korteweg-de Vries (mKdV) hierarchies via nonlocal symmetry reductions of matrix spectral problems was explored with their soliton solutions by the IST in [14]. Exact periodic and localized solutions of a nonlocal STR-Mel'nikov equation were derived by the HBM [15]. In [16], time reverse modeling was applied to localize and characterize acoustic emission using a numerical concrete model with a method for exploration of geophysics to nondestructive testing.…”

Multiwave interaction solutions of the partial reverse Space-time nonlocal Mel’nikov equation

“…Based on the Mel'nikov equation [27], researchers proposed the nonlocal Mel'nikov equation [28,29] where ò = ± 1, * represents complex conjugation. This equation describes the interaction of long wave with short-wave packets, and can be considered as a generalization of the Kadomtsev-Petviashvili (KP) equation [19].…”

“…Soliton and semi-rational solutions of equation (1) have been studied in [28] and the KP hierarchy reduction method is applied to construct Gram-type solutions of equation (1) in [30].…”