Prohibitions caused by nonlocality for nonlocal Boussinesq‐KdV type systems

Abstract

It is found that two different celebrate models, the Korteweg de-Vrise (KdV) equation and the Boussinesq equation, are linked to a same model equation but with different nonlocalities. The nonlocal KdV equation can be derived in two ways, via the so-called consistent correlated bang companied by the parity and time reversal from the local KdV equation and via the parity and time reversal symmetry K E Y W O R D S mathematical physics, nonlinear waves, solitons and integrable systems Stud Appl Math. 2019;143:123-138.wileyonlinelibrary.com/journal/sapm

Symmetry Breaking Soliton, Breather, and Lump Solutions of a Nonlocal Kadomtsev–Petviashvili System

“…Based on the bilinear form 12, we can first determine symmetry breaking soliton solutions through the Bäcklund transformation (10) of the AB-KP systems (7a) and (7b) with the function f being written as a summation of some special functions [11][12][13][14][15][16]18]:…”

“…Based on this, the revolutionary works, which named the Alice-Bob (AB) systems to describe two-place physical problems, were made by Lou recently [11,12]. e technical approach originated from the so-called P-T-C principle with P (the parity), T (time reversal), and C (charge conjugation) [11][12][13][14][15][16][17][18][19][20][21][22][23][24][25]. From this, a general Nth Darboux transformation for the AB-mKdV equation was constructed [13].…”

“…By using this Darboux transformation, some types of PT symmetry breaking solutions including soliton and rogue wave solutions were explicitly obtained. Combined with their Hirota bilinear forms, prohibitions caused by nonlocality for nonlocal Boussinesq-KdV type systems were investigated [14]. e two/four-place nonlocal Kadomtsev-Petviashvili (KP) equation were also explicitly solved for special types of multiple soliton solutions via a P-T-C symmetric-antisymmetric separation approach [15].…”

“…plays a crucial role in constructing analytical group invariant multisoliton solutions of the AB systems, including the KdV-KP-Toda type, mKdV-sG type, NLS type, and discrete H 1 type AB systems [11][12][13][14][15][16]18]. In this paper, we consider the KP equation 3as an illustrative example, which is one of the well-studied models of nonlinear waves in dispersive media [26,27] and in multicomponent plasmas [28].…”

Nonlocal-derivative NLS equations and group-invariant soliton solutions

“…systems have attracted considerable attention in recent years. At present, the nonlocal KdV equation [4,5], the nonlocal mKdV equation [6][7][8], the nonlocal discrete NLS equation [9], the nonlocal KP equation [10,11], the nonlocal DS equation [12][13][14], and so on [15][16][17] have been studied. Solitons represent robust nonlinear coherent structures and have been theoretically studied and observed in experiments in physical, chemical and biological science [18][19][20].…”