Darboux–Bäcklund transformations, dressing &amp; impurities in multi-component NLS

Adamopoulou, Panagiota; Doikou, Anastasia; Papamikos, Georgios

doi:10.1016/j.nuclphysb.2017.02.016

Cited by 12 publications

(16 citation statements)

References 25 publications

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“…Note that in earlier works [8,9,12] the Lax pairs as well as the general Darboux transforms for the vector and matrix NLS models were expressed as c-number matrices. Here we are offering a unifying dressing scheme regardless of the particular form of the operators.…”

Section: General Darboux Transform and Solutionsmentioning

confidence: 99%

Non-commutative NLS-type hierarchies: Dressing & solutions

2019

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We consider the generalized matrix non-linear Schrödinger (NLS) hierarchy. By employing the universal Darboux-dressing scheme we derive solutions for the hierarchy of integrable PDEs via solutions of the matrix Gelfand-Levitan-Marchenko equation, and we also identify recursion relations that yield the Lax pairs for the whole matrix NLS-type hierarchy. These results are obtained considering either matrix-integral or general n th order matrixdifferential operators as Darboux-dressing transformations. In this framework special links with the Airy and Burgers equations are also discussed. The matrix version of the Darboux transform is also examined leading to the non-commutative version of the Riccati equation. The non-commutative Riccati equation is solved and hence suitable conserved quantities are derived. In this context we also discuss the infinite dimensional case of the NLS matrix model as it provides a suitable candidate for a quantum version of the usual NLS model. Similarly, the non-commutitave Riccati equation for the general dressing transform is derived and it is naturally equivalent to the one emerging from the solution of the auxiliary linear problem.

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Section: General Darboux Transform and Solutionsmentioning

confidence: 99%

Non-commutative NLS-type hierarchies: Dressing & solutions

2019

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“…Proof. If the Darboux matrix (20) is written as the product of W (λ) and H(λ) which are elements of LG q for all λ, and of the form (25), then equation (21) takes the form (26), assuming that H(λ)JH(λ) −1 = J for all λ. It follows from (25) that…”

Section: The Vector Mkdv Hierarchymentioning

confidence: 99%

“…Moreover, in [21], efficient numerical integration schemes of (1) where considered. We note here that the vmKdV equation which is the third member of the vector nonlinear Schrödinger (vNLS) hierarchy (see for example [25]) is of the form u t + u xxx + 3 4 u T u x u + 3 4 u 2 u x = 0 , thus, different from the vmKdV equation that we consider in this paper. A relation between (1) and the nonlinear Schrödinger equation (NLS) is discussed in Section 3.…”

Section: Introductionmentioning

confidence: 95%

Drinfel'd-Sokolov construction and exact solutions of vector modified KdV hierarchy

Adamopoulou

Papamikos

2020

Nuclear Physics B

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We construct the hierarchy of a multi-component generalisation of modified KdV equation and find exact solutions to its associated members. The construction of the hierarchy and its conservation laws is based on the Drinfel'd-Sokolov scheme, however, in our case the Lax operator contains a constant non-regular element of the underlying Lie algebra. We also derive the associated recursion operator of the hierarchy using the symmetry structure of the Lax operators. Finally, using the rational dressing method, we obtain the one soliton solution, and we find the one breather solution of general rank in terms of determinants.

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“…This strategy seeks to reduce the approximation error. Substituting (18) into (17), we obtain the solution in function of , , and .…”

Section: Hpm Solutionmentioning

confidence: 99%

“…There are several methods of solutions for solving nonlinear differential equations such as inverse scattering transformation [14,15], Darboux transformation [16,17], bilinear method [18,19], the tanh-function method [20][21][22], the variable separation approach [23][24][25], the symmetry method [26,27], sine-cosine method [28][29][30], Adomian decomposition method (ADM) [12,[31][32][33], and homotopy perturbation method (HPM) [34][35][36]. HPM is based on the use of power series of , which transforms a differential equation into a set of linear differential equations.…”

Section: Introductionmentioning

confidence: 99%

Optimized Direct Padé and HPM for Solving Equation of Oxygen Diffusion in a Spherical Cell

Sandoval-Hernandez

Vázquez-Leal

Sarmiento-Reyes

et al. 2018

Discrete Dynamics in Nature and Society

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This work presents the application of homotopy perturbation method (HPM) and Optimized Direct Padé (ODP) to obtain a handy and easily computable approximate solution of the nonlinear differential equation to model the oxygen diffusion in a spherical cell with nonlinear oxygen uptake kinetics. On one hand, the obtained HPM solution is fully symbolic in terms of the coefficients of the equation, allowing us to use the same solution for different values of the maximum reaction rate, the Michaelis constant, and the permeability of the cell membrane. On the other hand, the numerical experiments show the high accuracy of the proposed ODP solution, resulting in 3.58×10-4 as the lowest absolute relative error (A.R.E.) for a set of coefficients. In addition, a novel technique is proposed to reduce the number of algebraic operations during the process of application of ODP method through the use of the Taylor series, which help to simplify the algebraic expressions used. The powerful process to obtain the solution shows that the Optimized Direct Padé and homotopy perturbation method are suitable methods to use.

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Darboux–Bäcklund transformations, dressing & impurities in multi-component NLS

Cited by 12 publications

References 25 publications

Non-commutative NLS-type hierarchies: Dressing & solutions

Non-commutative NLS-type hierarchies: Dressing & solutions

Drinfel'd-Sokolov construction and exact solutions of vector modified KdV hierarchy

Optimized Direct Padé and HPM for Solving Equation of Oxygen Diffusion in a Spherical Cell

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