Abstract The inverse scattering transform allows explicit construction of solutions to many physically significant nonlinear wave equations. Notably, this method can be extended to fractional nonlinear evolution equations characterized by anomalous dispersion using completeness of suitable eigenfunctions of the associated linear scattering problem. In anomalous diffusion, the mean squared displacement is proportional to t α , α > 0, while in anomalous dispersion, the speed of localized waves is proportional to A α , where A is the amplitude of the wave. Fractional extensions of the modified Korteweg–deVries (mKdV), sine-Gordon (sineG) and sinh-Gordon (sinhG) and associated hierarchies are obtained. Using symmetries present in the linear scattering problem, these equations can be connected with a scalar family of nonlinear evolution equations of which fractional mKdV (fmKdV), fractional sineG (fsineG), and fractional sinhG (fsinhG) are special cases. Completeness of solutions to the scalar problem is obtained and, from this, the nonlinear evolution equation is characterized in terms of a spectral expansion. In particular, fmKdV, fsineG, and fsinhG are explicitly written. One-soliton solutions are derived for fmKdV and fsineG using the inverse scattering transform and these solitons are shown to exhibit anomalous dispersion.
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This content will become publicly available on December 1, 2025
Characterizing the dispersion behavior of poly-atomic magnetic metamaterials
The propagation of magnetoinductive (MI) waves across magnetic metamaterials known as magnetoinductive waveguides (MIWs) has been an area of interest for many applications due to the flexible design and low-loss performance in challenging radio-frequency (RF) environments. Thus far, the dispersion behavior of MIWs has been limited to mono- and diatomic geometries. In this work, we present a recursive method to generate the dispersion equation for a general poly-atomic MIW. This recursive method greatly simplifies the ability to create closed-form dispersion equations for unique poly-atomic MIW geometries versus the previous method. To demonstrate, four MIW geometries that have been selected for their unique symmetries are analyzed using the recursive method. Using applicable simplifications brought on by the geometric symmetries, a closed-form dispersion equation is reported for each case. The equations are then validated numerically and show excellent agreement in all four cases. This work simultaneously aids in the further development of MIW theory and offers new avenues for MIW design in the presented dispersion equations.
more » « less- Award ID(s):
- 2053318
- PAR ID:
- 10524060
- Publisher / Repository:
- Nature
- Date Published:
- Journal Name:
- Scientific Reports
- Volume:
- 14
- Issue:
- 1
- ISSN:
- 2045-2322
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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