TY - JOUR

T1 - Nonlinear Fourier transform of time-limited and one-sided signals

AU - Vaibhav, Vishal

N1 - Accepted Author Manuscript

PY - 2018

Y1 - 2018

N2 - In this article, we study the properties of the nonlinear Fourier spectrum in order to gain better control of the temporal support of the signals synthesized using the inverse nonlinear Fourier transform. In particular, we provide necessary and sufficient conditions satisfied by the nonlinear Fourier spectrum such that the generated signal has a prescribed support. In our exposition, we assume that the support is a simply connected domain that is either a bounded interval or the half-line, which amounts to studying the class of signals which are either time-limited or one-sided, respectively. Further, it is shown that the analyticity properties of the scattering coefficients of the aforementioned classes of signals can be exploited to improve the numerical conditioning of the differential approach of inverse scattering. Here, we also revisit the integral approach of inverse scattering and provide the correct derivation of the so called Töplitz inner-bordering algorithm. Finally, we conduct extensive numerical tests in order to verify the analytical results presented in the article. These tests also provide us an opportunity to compare the performance of the two aforementioned numerical approaches in terms of accuracy and complexity of computations.

AB - In this article, we study the properties of the nonlinear Fourier spectrum in order to gain better control of the temporal support of the signals synthesized using the inverse nonlinear Fourier transform. In particular, we provide necessary and sufficient conditions satisfied by the nonlinear Fourier spectrum such that the generated signal has a prescribed support. In our exposition, we assume that the support is a simply connected domain that is either a bounded interval or the half-line, which amounts to studying the class of signals which are either time-limited or one-sided, respectively. Further, it is shown that the analyticity properties of the scattering coefficients of the aforementioned classes of signals can be exploited to improve the numerical conditioning of the differential approach of inverse scattering. Here, we also revisit the integral approach of inverse scattering and provide the correct derivation of the so called Töplitz inner-bordering algorithm. Finally, we conduct extensive numerical tests in order to verify the analytical results presented in the article. These tests also provide us an opportunity to compare the performance of the two aforementioned numerical approaches in terms of accuracy and complexity of computations.

KW - integrable nonlinear partial differential equation

KW - inverse scattering

KW - nonlinear optics

UR - http://resolver.tudelft.nl/uuid:66d9a7ea-8524-468e-912a-1f8114c087ba

UR - http://www.scopus.com/inward/record.url?scp=85053401705&partnerID=8YFLogxK

U2 - 10.1088/1751-8121/aad9ab

DO - 10.1088/1751-8121/aad9ab

M3 - Article

AN - SCOPUS:85053401705

VL - 51

JO - Journal of Physics A: Mathematical and Theoretical

JF - Journal of Physics A: Mathematical and Theoretical

SN - 1751-8113

IS - 42

M1 - 425201

ER -