Novel exact solutions of coupled nonlinear Schrödinger equations with time—space modulation

Abstract

We construct various novel exact solutions of two coupled dynamical nonlinear Schrödinger equations. Based on the similarity transformation, we reduce the coupled nonlinear Schrödinger equations with time-and space-dependent potentials, nonlinearities, and gain or loss to the coupled dynamical nonlinear Schrödinger equations. Some special types of non-travelling wave solutions, such as periodic, resonant, and quasiperiodically oscillating solitons, are used to exhibit the wave propagations by choosing some arbitrary functions. Our results show that the number of the localized wave of one component is always twice that of the other one. In addition, the stability analysis of the solutions is discussed numerically.

Exact solutions of the nonlinear differential—difference equations associated with the nonlinear electrical transmission line through a variable-coefficient discrete (G′/G)-expansion method

“…In particular, if C 1 = 0, C 2 = 0 then equations (33) and (34) can be written as the following solutions:…”

“…These are probably the reasons why, since pioneering works by Hirota and Suzuki [6] on electrical transmission lines simulating Toda lattice, a growing interest has been devoted to the use of nonlinear transmission lines, in particular, for studying nonlinear waves and nonlinear modulated waves: pulse solitons, envelope pulse (bright), hole (dark) solitons and kink and anti-kink solitons, [7][8][9] intrinsic localized modes (also called discrete breathers), [10][11][12] modulational instability. [13][14][15][16][17] Several methods for finding the exact solutions of nonlinear evolution equations for mathematical physics have been proposed, [18][19][20][21][22][23][24][25][26][27][28][29][30][31][32][33][34] such as Jacobian elliptic function method, [35,36] Fibonacci tanh function for NDDEs, [37] Expfunction method, [38,39] variable-coefficient discrete (G /G)expansion method for NDDEs, [40] and so on. In order to establish the effectiveness and reliability of the (G /G)-expansion method and to expand the possibility of its application, further research has been carried out by a considerable number of researchers.…”

Spatial weak-light ring soliton in self-assembled quantum dots

“…It is observed that the two-dimensional spatial optical soliton obtained is stable. [17] Since the soliton may propagate stably, we then study the dynamical properties. To be exact to the first order, the soliton solution of Eq.…”

“…[5][6][7][8][9][10][11][12][13] So, the spatial optical solitons in EIT media, such as waveguide, atomic media, and quantum well, have received increasing attention. [14][15][16][17][18][19][20][21][22][23][24][25][26][27] Among these media, the atomic system plays a leading role in studying the spatial optical pulse with vanishing absorption solitons. However, one potential shortcoming of the atomic system is that it cannot be conveniently used in devices with good scalability.…”

(2+1)-dimensional dissipation nonlinear Schrödinger equation for envelope Rossby solitary waves and chirp effect

“…Besides the above-mentioned problems, we shall use these methods to solve the soliton equations, such as the first integral method, [19] similarity transformation, [20] Jacobi elliptic function expansion method, [21] Bäcklund transformation method, [22] Darboux transformation method, [23] and so on. In the paper, we will derive the one-soliton and two-soliton solutions of the (2+1)-dimensional NLS equation by employing the Hirota method [24] and analyze the chirp effect based on the analytical solutions.…”