Modulation instability, conservation laws and soliton solutions for an inhomogeneous discrete nonlinear Schrödinger equation

New Optical Solitons of the Longitudinal Wave Equation in a Magnetoelectro-Elastic Circular Rod

“…Zayed and Alurrfi [22] reported the results of a research in which the modified Kudryashov method was adopted to produce a number of exact solutions for nonlinear seventh-order SawadaKotera-Ito equation, the nonlinear seventh-order KaupKupershmidt equation and the nonlinear seventh-order Lax equation. For more research papers, see [23][24][25][26][27][28][29][30][31][32][33][34][35][36][37][38][39][40][41][42]. This paper illustrates the effectiveness of the modified Kudryashov method in finding a number of new optical solitons for the nonlinear longitudinal wave equation in * corresponding author; e-mail: qinzhou@whu.edu.cn a magnetoelectro-elastic circular rod [43] …”

Dynamics of higher‐order rational and semi‐rational soliton solutions of the coupled modified KdV lattice equation

“…Figure 24 exhibits the third-order RS solutions (38) with N = 3 of Equation ( 1), time evolutions of using the third-order RS solutions, and those perturbed by FIGURE 24 (Color online). Third-order RS solutions (38): (A1,A2) exact solution (38) with e 2 = 10000, e 1 = d 1 = d 2 = 0; (B1,B2) time evolution of the wave using exact solution (38) as the initial condition; (C1,C2) time evolution using exact solution (38) perturbed by a 2% noise as the initial condition [Colour figure can be viewed at wileyonlinelibrary.com] a small noise 2% with the same parameters as in Figure 23. It is clearly seen that our numerical solutions almost exactly reproduce the analytical solutions; in other words, these solutions have the stable evolutions without a noise, which also shows the accuracy of our numerical scheme.…”

“…With the aid of symbolic computation, the third-order RS solutions can be explicitly deduced by (38) which are of the complicated expressions about n, t and parameters e 1 , d 1 , e 2 , and d 2 and omitted here. The profiles of the third-order RS solutions (38) are displayed in Figures 19-23. Actually, the parameters e i , d i (i = 1, 2) can split the third-order RSs (38) into three parallel solitons which are exhibited in Figures 21 and 23.…”

“…Actually, the parameters e i , d i (i = 1, 2) can split the third-order RSs (38) into three parallel solitons which are exhibited in Figures 21 and 23. Similarly, for the third-order RS solutions (38), if we take the transformations n → n + 𝜗, t → t + 𝜅, then the controllable third-order RS solutions will be given; here, we don't discuss them. The parameters e i , d i (i = 1, 2) excite the third-order RS solutions (38)to generate the abundant wave structures.…”

“…To understand such phenomena described by the NDDEs, it is significant to study the exact soliton solutions of the NDDEs. Several methods have been established to find the soliton solutions, including the inverse scattering transform method, 28,29 the Hirota bilinear method, 30 the algebra‐geometric technique, 31 and the Darboux transformation (DT) method 32–40 . Among them, the DT method is known to be powerful in finding soliton solutions of NDDEs.…”