Bäcklund Transformations and Exact Soliton Solutions for Some Nonlinear Evolution Equations of the zs/akns Systemfn1fn1Communicated by Prof. M. Wadati.

Quasideterminant solutions of NC Painlevé II equation with the Toda solution at n=1 as a seed solution in its Darboux transformation

“…Further, I derived a zero curvature representation of quantum Painlevé II equation with its associated Riccati form and also we have derived an explicit expression of NC PII Riccati equation from the linear system of NC PII equation by using the method of Konno and Wadati [ [28]]. Further, one can derive Bcklund transformations for NC PII equation with the help of our NC PII linear system and its Riccati form by using the technique described in [28,29], these transformations may be helpful to construct the nonlinear principle of superposition for NC PII solutions. It also seems interesting to construct the connection of NC PII equation to the known integrable systems such as its connection to NC nonlinear Schrödinger equation and to NC KdV equation as it possesses this property in classical case.…”

Construction of solitary wave solutions for variants of theK(n, n) equation

“…To solve the problem using the decomposition method, we simply substitute Equation (17) into Equation (14), to obtain the following recurrence relation…”

“…Explicit solutions to such equations are of fundamental importance and there is a strong interest in explicit soliton solutions. Solitons are found in many physical phenomena, as they arise as the solutions of a widespread class of weakly nonlinear dispersive partial differential equations describing physical systems [10,[12][13][14][19][20][21][22][23][25][26][27][29][30][31][32][33][34][35][36]38,42]. The soliton concept has been studied by many analytical and numerical methods such as the inverse scattering method, Backlund transformation method, the pseudo spectral method [21], the tri-Hamiltonian operators [19], the finite difference method [10], the Adomian decomposition method (ADM) [25], the tanh-coth method [36], the sine-cosine and tanh methods [31][32][33][34][35].…”

A conservative numerical method for a class of unstable nonlinear Schrödinger equation

“…Recently, increasing attention has also been paid to analysis of nonlinear pulse propagation in optical fibers described by NLS equation, especially the study of the soliton phenomenon in unstable media [2][3][4][5][6][7][8][9][10][11]. In order to describe the nonlinear wave's propagation in unstable media, the authors in Ref.…”