Intertwining technique for the matrix Schrödinger equation

Travelling wave solutions and soliton solutions for the nonlinear transmission line using the generalized Riccati equation mapping method

“…(16)(17)(18)(19)(20)(21) and using the computer programs such as Maple or MATLAB, we obtain the following results:…”

“…In the literature, many of these methods among which we find are: the Hirotas bilinear method [19], the Backlund transformation method [20], the Darboux transformation method [21], the Painleve singularity structure analysis method [22], the Riccati expansion with constant coefficients [23], the variational iteration method [24], the exp-function method [25], the algebraic method [26], the collocation method [27], the Kudryashov method [28][29][30][31][32], the (G /G)-expansion method [33][34][35][36][37][38], the simplest equation method [39][40][41][42][43], and so on. However, some of these analytical methods are not easy to handle and are often subject to tedious mathematical developments.…”

Intertwining relations and Darboux transformations for the wave equations

“…The solution χ 2 can be obtained by applying the oper ator ᏸ = ᏸ 2 ᏸ 1 to the transformation solution ᐁ 3 at energy λ 3 , that is According to (65), the solution χ 2 can be written as [14]. It would be also interesting to generate Darboux trans formations for the generalized Schrödinger equations by using associated functions.…”

“…The technique of iterated integral operators for Schrödinger equation is based on the integral Gel fand-Levitan and Marchenko equations [60][61][62]. For example, the Darboux transformations in the integral form were considered for the conventional Schrödinger equation in [63,64], the Darboux trans formations in the differential and in the integral form and its relationships were studied for the system of Schrödinger equations in [65], and for the Schrödinger equation with variable energy and angular momentum in [66,67]. As we know, the first attempts to construct the second order Darboux transformations in the inte gral form for the Schrödinger equation with position dependent mass and weighted energy was made in [68].…”

Minimal realizations of supersymmetry for matrix Hamiltonians

“…The matrix models with supersymmetry appear in Quantum Mechanics in several areas: in particular, for spectral design of potentials describing multichannel scattering and the motion of spin particles in external fields. The different cases of such models are considered in [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16] and their systematic study was undertaken in [17][18][19][20][21][22][23][24][25][26][27][28][29][30][31][32][33] (see also the recent reviews [34,35]). In [17] intertwining of matrix Hermitian Hamiltonians by n × n first-order and 2 × 2 second-order matrix differential operators was investigated and the corresponding supersymmetric algebras were constructed.…”