scholarly journals Solitons in lattice field theories via tight-binding supersymmetry

2021 ◽  
Vol 2021 (7) ◽  
Author(s):  
Shankar Balasubramanian ◽  
Abu Patoary ◽  
Victor Galitski
Keyword(s):  
Inverse Scattering ◽  
Continuum Limit ◽  
Scattering Problem ◽  
Tight Binding ◽  
Soliton Solutions ◽  
Inverse Scattering Method ◽  
Field Theories ◽  
Scattering Method ◽  
De Vries ◽  
The Continuum

Abstract Reflectionless potentials play an important role in constructing exact solutions to classical dynamical systems (such as the Korteweg-de Vries equation), non-perturbative solutions of various large-N field theories (such as the Gross-Neveu model), and closely related solitonic solutions to the Bogoliubov-de Gennes equations in the theory of superconductivity. These solutions rely on the inverse scattering method, which reduces these seemingly unrelated problems to identifying reflectionless potentials of an auxiliary one-dimensional quantum scattering problem. There are several ways of constructing these potentials, one of which is quantum mechanical supersymmetry (SUSY). In this paper, motivated by recent experimental platforms, we generalize this framework to develop a theory of lattice solitons. We first briefly review the classical inverse scattering method in the continuum limit, focusing on the Korteweg-de Vries (KdV) equation and SU(N) Gross-Neveu model in the large N limit. We then generalize this methodology to lattice versions of interacting field theories. Our analysis hinges on the use of trace identities, which are relations connecting the potential of an equation of motion to the scattering data. For a discrete Schrödinger operator, such trace identities had been known as far back as Toda; however, we derive a new set of identities for the discrete Dirac operator. We then use these identities in a lattice Gross-Neveu and chiral Gross-Neveu (Nambu-Jona-Lasinio) model to show that lattice solitons correspond to reflectionless potentials associated with the discrete scattering problem. These models are of significance as they are equivalent to a mean-field theory of a lattice superconductor. To explicitly construct these solitons, we generalize supersymmetric quantum mechanics to tight-binding models. We show that a matrix transformation exists that maps a tight-binding model to an isospectral one which shares the same structure and scattering properties. The corresponding soliton solutions have both modulated hopping and onsite potential, the former of which has no analogue in the continuum limit. We explicitly compute both topological and non-topological soliton solutions as well as bound state spectra in the aforementioned models.

SUPERSYMMETRY, REFLECTIONLESS SYMMETRIC POTENTIALS AND THE INVERSE METHOD

1990 ◽  
Vol 05 (09) ◽  
pp. 1763-1772 ◽  
Author(s):  
B. BAGCHI
Keyword(s):  
Inverse Scattering ◽  
Schrodinger Equation ◽  
Inverse Method ◽  
Kdv Equation ◽  
Soliton Solutions ◽  
Inverse Scattering Method ◽  
Wave Functions ◽  
Scattering Method ◽  
De Vries

The role of inverse scattering method is illustrated to examine the connection between the multi-soliton solutions of Korteweg-de Vries (KdV) equation and discrete eigenvalues of Schrödinger equation. The necessity of normalization of the Schrödinger wave functions, which are constructed purely from a supersymmetric consideration is pointed out.

Discrete inverse methods for elastic waves in layered media

Geophysics ◽  
1980 ◽  
Vol 45 (2) ◽  
pp. 213-233 ◽  
Author(s):  
James G. Berryman ◽  
Robert R. Greene
Keyword(s):  
Inverse Problem ◽  
Wave Equation ◽  
Finite Number ◽  
Inverse Scattering ◽  
Continuum Limit ◽  
Schroedinger Equation ◽  
Inverse Scattering Method ◽  
Inverse Methods ◽  
Scattering Method ◽  
The Continuum

The seismic inverse problem for waves at normal incidence on horizontally layered media is discussed. The emphasis is theoretical rather than practical, but some long‐standing questions concerning the general applicability of the often taught Goupillaud inverse method are answered. The main purpose is to demonstrate in detail the equivalence between the Goupillaud method of inversion for the wave equation and the Marchenko integral equation (inverse scattering) method for the Schroedinger equation. We show that the very simple method of solution due to Goupillaud for a specialized model (layers of equal traveltime) actually has a much wider significance. If seismic data are smoothed before sampling using a type of antialiasing filter, the Goupillaud method gives a valid approximate inversion for models with arbitrary layer thicknesses (or continuous impedance variation) when the “reflection coefficients” are appropriately reinterpreted. In all, three inverse methods are considered: (1) the Goupillaud method for the wave equation and both (2) continuous and (3) discrete inverse scattering methods for the Schroedinger equation. A computationally fast algorithm for solving the inverse scattering formulas is deduced from the equivalent Goupillaud method. By comparing the continuous and discrete formalisms in the continuum limit, a preferred form is found within the class of symmetric tridiagonal discretizations of the Schroedinger equation. For the elastic wave inverse problem, two cases are distinguished: (1) If the impedance is continuous, we show that both the Goupillaud method and the discrete inverse scattering method converge to the impedance when the equal‐traveltime layer thickness goes to zero; and (2) if the impedance has a finite number of discontinuities, we show that the inverse scattering method assigns the arithmetic average across the discontinuity at the point of discontinuity, while the Goupillaud method assigns the value of the right‐hand (spatially deeper) limit. Thus, in the continuum limit, both methods will reconstruct the same impedance except (possibly) for the values at a finite number of jump points in any finite span of traveltime.

EXACT SOLUTIONS OF NONLINEAR SU(N) PRINCIPAL σ-MODELS

1988 ◽  
Vol 03 (05) ◽  
pp. 1147-1154
Author(s):  
TIBOR KISS-TOTH
Keyword(s):  
Exact Solutions ◽  
Inverse Scattering ◽  
Soliton Solutions ◽  
Static Solution ◽  
Finite Energy ◽  
Single Step ◽  
Inverse Scattering Method ◽  
Scattering Method ◽  
Axially Symmetric ◽  
Symmetric Static Solution

The superpotential for n-step soliton solution is derived in the case of an arbitrary dimensional projector for axially symmetric, static solution of nonlinear principal SU (N) σ-models. This was done by using an inverse scattering method developed by Belinski and Zakharov. Finite energy solutions are constructed for all SU (N) one soliton solutions generated by a single step.

Double-Alekseev inverse scattering method in the stationary axi-symmetric vacuum gravitation field equations

Open Physics ◽  
2008 ◽  
Vol 6 (3) ◽  
Author(s):  
Chun-Yan Wang ◽  
Yuan-Xing Gui ◽  
Ya-Jun Gao
Keyword(s):  
Inverse Scattering ◽  
Function Method ◽  
Complex Function ◽  
Soliton Solutions ◽  
Inverse Scattering Method ◽  
Scattering Method ◽  
Field Equations ◽  
Double Complex ◽  
Gravitation Field ◽  
Symmetric Vacuum

AbstractWe present a new improvement to the Alekseev inverse scattering method. This improved inverse scattering method is extended to a double form, followed by the generation of some new solutions of the double-complex Kinnersley equations. As the double-complex function method contains the Kramer-Neugebauer substitution and analytic continuation, a pair of real gravitation soliton solutions of the Einstein’s field equations can be obtained from a double N-soliton solution. In the case of the flat Minkowski space background solution, the general formulas of the new solutions are presented.

Explicit N -Soliton solutions in the Inverse Scattering Method

1989 ◽  
Vol 12 (3) ◽  
pp. 327-332 ◽  
Author(s):  
Chen Zong–yun ◽  
Huang Nian–ning ◽  
Xiao Yi
Keyword(s):  
Inverse Scattering ◽  
Soliton Solutions ◽  
Inverse Scattering Method ◽  
Scattering Method

Polynomial conserved densities for the sine-Gordon equations

1977 ◽  
Vol 352 (1671) ◽  
pp. 481-503 ◽  
Keyword(s):  
Inverse Scattering ◽  
Wave Equations ◽  
Gordon Equation ◽  
Numerical Solutions ◽  
Soliton Solutions ◽  
Inverse Scattering Method ◽  
Bäcklund Transformations ◽  
Scattering Method ◽  
Sine Gordon Equation ◽  
Backlund Transformations

Like a number of other nonlinear dispersive wave equations the sine–Gordonequation z , xt = sin z has both multi-soliton solutions and an infinity of conserved densities which are polynomials in z , x , z , xx , etc. We prove that the generalized sine–Gordon equation z , xt = F ( z ) has an infinity of such polynomial conserved densities if, and only if, F ( z ) = A e αz + B e – αz for complex valued A, B and α ≠ 0. If F ( z ) does not take the form A e αz + B e βz there is no p. c. d. of rank greater than two. If α ≠ – β there is only a finite number of p. c. ds. If α = – β then if A and B are non-zero all p. c. ds are of even rank; if either A or B vanishes the p. c. ds are of both even and odd ranks. We exhibit the first eleven p. c. ds in each case when α = – β and the first eight when α ≠ – β . Neither the odd rank p. c. ds in the case α = – β , nor the particular limited set of p. c. ds in the case when α ≠ – β have been reported before. We connect the existence of an infinity of p. c. ds with solutions of the equations through an inverse scattering method, with Bäcklund transformations and, via Noether’s theorem, with infinitesimal Bäcklund transformations. All equations with Bäcklund transformations have an infinity of p. c. ds but not all such p. c. ds can be generated from the Bäcklund transformations. We deduce that multiple sine–Gordon equations like z , xt = sin z + ½ sin ½ z , which have applications in the theory of short optical pulse propagation, do not have an infinity of p. c. ds. For these equations we find essentially three conservation laws: one and only one of these is a p. c. d. and this is of rank two. We conclude that the multiple sine–Gordons will not be soluble by present formulations of the inverse scattering method despite numerical solutions which show soliton like behaviour. Results and conclusions are wholly consistent with the theorem that the generalized sine–Gordon equation has auto-Bäcklund transformations if, and only if Ḟ ( z ) – α 2 F ( z ) = 0.

scholarly journals SEMI-ANALYTIC EQUATIONS TO THE COX–THOMPSON INVERSE SCATTERING METHOD AT FIXED ENERGY FOR SPECIAL CASES

2008 ◽  
Vol 22 (23) ◽  
pp. 2191-2199 ◽  
Author(s):  
TAMÁS PÁLMAI ◽  
MIKLÓS HORVÁTH ◽  
BARNABÁS APAGYI
Keyword(s):  
Inverse Scattering ◽  
Scattering Problem ◽  
Identical Particle ◽  
Inverse Scattering Problem ◽  
Inverse Scattering Method ◽  
Scattering Method ◽  
Fixed Energy ◽  
Angular Momenta ◽  
Partial Waves ◽  
Special Cases

Solution of the Cox–Thompson inverse scattering problem at fixed energy1–3 is reformulated resulting in semi-analytic equations. The new set of equations for the normalization constants and the nonphysical (shifted) angular momenta are free of matrix inversion operations. This simplification is a result of treating only the input phase shifts of partial waves of a given parity. Therefore, the proposed method can be applied for identical particle scattering of the bosonic type (or for certain cases of identical fermionic scattering). The new formulae are expected to be numerically more efficient than the previous ones. Based on the semi-analytic equations an approximate method is proposed for the generic inverse scattering problem, when partial waves of arbitrary parity are considered.

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