Reverse Space‐Time Nonlocal Sine‐Gordon/Sinh‐Gordon Equations with Nonzero Boundary Conditions

Abstract

Nonlocal reverse space-time Sine/Sinh-Gordon type equations were recently introduced. They arise from a remarkably simple nonlocal reduction of the well-known AKNS scattering problem, hence, they constitute an integrable evolution equations. Furthermore, the inverse scattering transform (IST) for rapidly decaying data was also constructed. In this paper, the IST for these novel nonlocal equations corresponding to nonzero boundary conditions (NZBCs) at infinity is presented. The NZBC problem is more complex due to the intricate branching structure of the associated linear eigenfunctions. Two cases are analyzed, which correspond to two different values of the phase at infinity. Special soliton solutions are discussed and explicit 1soliton and 2-soliton solutions are found. Both spatially independent and spatially dependent boundary conditions are considered.

Discrete symmetries and nonlocal reductions

“…where ǫ 1 = ǫ 2 = ±1 and k 2 = 1. If we take q ′ = q and r ′ = r these special discrete transformations produce the nonlocal reductions: r(t, x) = kq(−t, x), r(t, x) = kq(t, −x), and r(t, x) = kq(−t, −x) [2]- [4], [19].…”

“…After Ablowitz and Musslimani's works there is a huge interest in obtaining nonlocal reductions of systems of integrable equations and finding interesting wave solutions of these systems. Specific examples are nonlocal NLS equation [1]- [14], nonlocal mKdV equation [2]- [4], [13], [15]- [18], nonlocal SG equation [2]- [4], [19], nonlocal DS equation [3], [20]- [24], nonlocal Fordy-Kulish equations [13], [25], nonlocal N -wave systems [3], [26], nonlocal vector NLS equations [27]- [30], nonlocal (2 + 1)dimensional negative AKNS systems [31], nonlocal coupled Hirota-Iwao mKdV systems [32]. See [33] for the discussion of superposition of nonlocal integrable equations, and [34] for the nonlocal reductions of the integrable equations of hydrodynamic type.…”

“…This means that the reduced systems admit recursion operators and Lax pairs. We can obtain N -soliton solutions of the reduced systems by the inverse scattering method [1]- [3], [10], [11], [14], [17], [19], [27], by the Darboux transformation [9], [16], [18], [22], [23], and by the Hirota bilinear method [7], [13], [15], [21], [31]- [33].…”

Nonlocal KdV equations

“…After Ablowitz and Musslimani introduced nonlocal type of reductions [1][2][3] we witnessed several works about finding new integrable nonlocal nonlinear partial differential equations and obtaining different kinds of solutions of these local and nonlocal equations. In particular, most of the works focused on nonlocal nonlinear Schrödinger equations (NLS) [1][2][3][4][5][6][7][8][9][10][11][12][13][14], nonlocal modified Korteweg-de Vries (mKdV) equations [2][3][4], [8], [15][16][17][18], nonlocal sine-Gordon (SG) equations [2][3][4], [19], and so on [20][21][22][23][24][25][26][27][28][29][30][31][32][33].…”

Physically significant nonlocal nonlinear Schrödinger equation and its soliton solutions

“…Following the introduction of this equation, its properties have been extensively investigated [6,[8][9][10][11][12][13][14][15][16][17][18][19]. In addition, other nonlocal integrable equations have been reported [20][21][22][23][24][25][26][27][28][29][30][31][32][33][34][35][36][37][38]. A transformation between many nonlocal and local equations has been discovered as well [34].…”