Fractional Integrable Nonlinear Soliton Equations

• URL: http://arxiv.org/abs/2203.13734v3
• Date: Wed, 20 Apr 2022 20:05:37 GMT
• Title: Fractional Integrable Nonlinear Soliton Equations
• Authors: Mark J. Ablowitz, Joel B. Been, Lincoln D. Carr
• Abstract summary: We present a new class of integrable fractional nonlinear evolution equations describing dispersive transport in fractional media. These equations can be constructed from nonlinear integrable equations using a widely generalizable mathematical process. We show that these equations are shown to predict super-dispersive transport of non-dissipative solitons in fractional media.
• Score: 0.0
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: Nonlinear integrable equations serve as a foundation for nonlinear dynamics, and fractional equations are well known in anomalous diffusion. We connect these two fields by presenting the discovery of a new class of integrable fractional nonlinear evolution equations describing dispersive transport in fractional media. These equations can be constructed from nonlinear integrable equations using a widely generalizable mathematical process utilizing completeness relations, dispersion relations, and inverse scattering transform techniques. As examples, this general method is used to characterize fractional extensions to two physically relevant, pervasive integrable nonlinear equations: the Korteweg-de Vries and nonlinear Schr\"odinger equations. These equations are shown to predict super-dispersive transport of non-dissipative solitons in fractional media.

Related papers

• Generalized and new solutions of the NRT nonlinear Schrödinger equation [0.0]
  We present new solutions of the non-linear Schr"oodinger equation proposed by Nobre, Rego-Monteiro and Tsallis for the free particle. Analytical expressions for the wave function, the auxiliary field and the probability density are derived using a variety of approaches.
  arXiv  Detail & Related papers  (2024-10-26T17:02:33Z)
• Nonlinear functionals of master equation unravelings [0.0]
  The trajectories generated by unravelings may also be treated as real -- as in the collapse models. Two types of nonlinear functionals are considered here: variance, and entropy. In the case of entropy, these corrections are shown to be negative, expressing the localization introduced by the Lindblad operators.
  arXiv  Detail & Related papers  (2024-02-08T02:21:23Z)
• Towards true discovery of the differential equations [57.089645396998506]
  Differential equation discovery is a machine learning subfield used to develop interpretable models. This paper explores the prerequisites and tools for independent equation discovery without expert input.
  arXiv  Detail & Related papers  (2023-08-09T12:03:12Z)
• Dynamical chaos in nonlinear Schr\"odinger models with subquadratic power nonlinearity [137.6408511310322]
  We deal with a class of nonlinear Schr"odinger lattices with random potential and subquadratic power nonlinearity. We show that the spreading process is subdiffusive and has complex microscopic organization. The limit of quadratic power nonlinearity is also discussed and shown to result in a delocalization border.
  arXiv  Detail & Related papers  (2023-01-20T16:45:36Z)
• Fractional Integrable and Related Discrete Nonlinear Schr\"odinger Equations [0.0]
  Integrable fractional equations such as the fractional Korteweg-deVries and nonlinear Schr"odinger equations are key to the intersection of nonlinear dynamics and fractional calculus. The first discrete/differential difference equation of this type is found, the fractional integrable discrete nonlinear Schr"odinger equation.
  arXiv  Detail & Related papers  (2022-10-03T21:03:22Z)
• D-CIPHER: Discovery of Closed-form Partial Differential Equations [80.46395274587098]
  We propose D-CIPHER, which is robust to measurement artifacts and can uncover a new and very general class of differential equations. We further design a novel optimization procedure, CoLLie, to help D-CIPHER search through this class efficiently.
  arXiv  Detail & Related papers  (2022-06-21T17:59:20Z)
• Exact solutions of interacting dissipative systems via weak symmetries [77.34726150561087]
  We analytically diagonalize the Liouvillian of a class Markovian dissipative systems with arbitrary strong interactions or nonlinearity. This enables an exact description of the full dynamics and dissipative spectrum. Our method is applicable to a variety of other systems, and could provide a powerful new tool for the study of complex driven-dissipative quantum systems.
  arXiv  Detail & Related papers  (2021-09-27T17:45:42Z)
• Nonlinear Independent Component Analysis for Continuous-Time Signals [85.59763606620938]
  We study the classical problem of recovering a multidimensional source process from observations of mixtures of this process. We show that this recovery is possible for many popular models of processes (up to order and monotone scaling of their coordinates) if the mixture is given by a sufficiently differentiable, invertible function.
  arXiv  Detail & Related papers  (2021-02-04T20:28:44Z)
• The Connection between Discrete- and Continuous-Time Descriptions of Gaussian Continuous Processes [60.35125735474386]
  We show that discretizations yielding consistent estimators have the property of invariance under coarse-graining' This result explains why combining differencing schemes for derivatives reconstruction and local-in-time inference approaches does not work for time series analysis of second or higher order differential equations.
  arXiv  Detail & Related papers  (2021-01-16T17:11:02Z)
• Data-driven peakon and periodic peakon travelling wave solutions of some nonlinear dispersive equations via deep learning [7.400475825464313]
  We apply the multi-layer physics-informed neural networks (PINNs) deep learning to study the data-driven peakon and periodic peakon solutions of some well-known nonlinear dispersion equations. Results will be useful to further study the peakon solutions and corresponding experimental design of nonlinear dispersive equations.
  arXiv  Detail & Related papers  (2021-01-12T09:50:28Z)
• Linear and integrable nonlinear evolution of the qutrit [0.0]
  The analyzed dynamics of the qutrit is rich and includes quasiperiodic motion, multiple equilibria and limit cycles. The generalization of the von Neumann equation preserving convexity of the state space is studied in the nontrivial case of the qutrit.
  arXiv  Detail & Related papers  (2020-06-18T07:25:19Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.