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a – e Conventional solitons can form in the presence of quadratic dispersion. a Equivalence of the dispersion operator in the time and frequency domains. b The dispersion (red) and (inverse) group velocity (dashed green) vary smoothly with frequency ω . The dot dashed vertical line marks the soliton central frequency ω 0 . c At low intensities I , dispersion stretches pulses in time. Solitons can form at high intensities with smooth ( d ) spectral and ( e ) temporal profiles that decay exponentially. f Corresponding dispersion operator for Hilbert-NLS solitons. g The dispersion relation Eq. has a discontinuous (disc.) <t>derivative</t> (red) and associated discontinuous (inverse) group velocity (dashed green) as a function of frequency; h At low intensities, the dispersion causes input pulses to split in two. At high intensities, solitons form, which have ( i ) a spectrum with discontinuous derivative, and ( k ) non-exponential decay in time. The units are arbitrary, and the intensity profiles in ( d , e , i ) and ( k ) are on logarithmic scales.
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a – e Conventional solitons can form in the presence of quadratic dispersion. a Equivalence of the dispersion operator in the time and frequency domains. b The dispersion (red) and (inverse) group velocity (dashed green) vary smoothly with frequency ω . The dot dashed vertical line marks the soliton central frequency ω 0 . c At low intensities I , dispersion stretches pulses in time. Solitons can form at high intensities with smooth ( d ) spectral and ( e ) temporal profiles that decay exponentially. f Corresponding dispersion operator for Hilbert-NLS solitons. g The dispersion relation Eq. has a discontinuous (disc.) <t>derivative</t> (red) and associated discontinuous (inverse) group velocity (dashed green) as a function of frequency; h At low intensities, the dispersion causes input pulses to split in two. At high intensities, solitons form, which have ( i ) a spectrum with discontinuous derivative, and ( k ) non-exponential decay in time. The units are arbitrary, and the intensity profiles in ( d , e , i ) and ( k ) are on logarithmic scales.
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molecule second mitochondria derived activator  (Mimetics)
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a – e Conventional solitons can form in the presence of quadratic dispersion. a Equivalence of the dispersion operator in the time and frequency domains. b The dispersion (red) and (inverse) group velocity (dashed green) vary smoothly with frequency ω . The dot dashed vertical line marks the soliton central frequency ω 0 . c At low intensities I , dispersion stretches pulses in time. Solitons can form at high intensities with smooth ( d ) spectral and ( e ) temporal profiles that decay exponentially. f Corresponding dispersion operator for Hilbert-NLS solitons. g The dispersion relation Eq. has a discontinuous (disc.) <t>derivative</t> (red) and associated discontinuous (inverse) group velocity (dashed green) as a function of frequency; h At low intensities, the dispersion causes input pulses to split in two. At high intensities, solitons form, which have ( i ) a spectrum with discontinuous derivative, and ( k ) non-exponential decay in time. The units are arbitrary, and the intensity profiles in ( d , e , i ) and ( k ) are on logarithmic scales.
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a – e Conventional solitons can form in the presence of quadratic dispersion. a Equivalence of the dispersion operator in the time and frequency domains. b The dispersion (red) and (inverse) group velocity (dashed green) vary smoothly with frequency ω . The dot dashed vertical line marks the soliton central frequency ω 0 . c At low intensities I , dispersion stretches pulses in time. Solitons can form at high intensities with smooth ( d ) spectral and ( e ) temporal profiles that decay exponentially. f Corresponding dispersion operator for Hilbert-NLS solitons. g The dispersion relation Eq. has a discontinuous (disc.) <t>derivative</t> (red) and associated discontinuous (inverse) group velocity (dashed green) as a function of frequency; h At low intensities, the dispersion causes input pulses to split in two. At high intensities, solitons form, which have ( i ) a spectrum with discontinuous derivative, and ( k ) non-exponential decay in time. The units are arbitrary, and the intensity profiles in ( d , e , i ) and ( k ) are on logarithmic scales.
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second mitochondria-derived activator of caspase prosmac  (Mimetics)
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a – e Conventional solitons can form in the presence of quadratic dispersion. a Equivalence of the dispersion operator in the time and frequency domains. b The dispersion (red) and (inverse) group velocity (dashed green) vary smoothly with frequency ω . The dot dashed vertical line marks the soliton central frequency ω 0 . c At low intensities I , dispersion stretches pulses in time. Solitons can form at high intensities with smooth ( d ) spectral and ( e ) temporal profiles that decay exponentially. f Corresponding dispersion operator for Hilbert-NLS solitons. g The dispersion relation Eq. has a discontinuous (disc.) <t>derivative</t> (red) and associated discontinuous (inverse) group velocity (dashed green) as a function of frequency; h At low intensities, the dispersion causes input pulses to split in two. At high intensities, solitons form, which have ( i ) a spectrum with discontinuous derivative, and ( k ) non-exponential decay in time. The units are arbitrary, and the intensity profiles in ( d , e , i ) and ( k ) are on logarithmic scales.
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a – e Conventional solitons can form in the presence of quadratic dispersion. a Equivalence of the dispersion operator in the time and frequency domains. b The dispersion (red) and (inverse) group velocity (dashed green) vary smoothly with frequency ω . The dot dashed vertical line marks the soliton central frequency ω 0 . c At low intensities I , dispersion stretches pulses in time. Solitons can form at high intensities with smooth ( d ) spectral and ( e ) temporal profiles that decay exponentially. f Corresponding dispersion operator for Hilbert-NLS solitons. g The dispersion relation Eq. has a discontinuous (disc.) <t>derivative</t> (red) and associated discontinuous (inverse) group velocity (dashed green) as a function of frequency; h At low intensities, the dispersion causes input pulses to split in two. At high intensities, solitons form, which have ( i ) a spectrum with discontinuous derivative, and ( k ) non-exponential decay in time. The units are arbitrary, and the intensity profiles in ( d , e , i ) and ( k ) are on logarithmic scales.
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a – e Conventional solitons can form in the presence of quadratic dispersion. a Equivalence of the dispersion operator in the time and frequency domains. b The dispersion (red) and (inverse) group velocity (dashed green) vary smoothly with frequency ω . The dot dashed vertical line marks the soliton central frequency ω 0 . c At low intensities I , dispersion stretches pulses in time. Solitons can form at high intensities with smooth ( d ) spectral and ( e ) temporal profiles that decay exponentially. f Corresponding dispersion operator for Hilbert-NLS solitons. g The dispersion relation Eq. has a discontinuous (disc.) <t>derivative</t> (red) and associated discontinuous (inverse) group velocity (dashed green) as a function of frequency; h At low intensities, the dispersion causes input pulses to split in two. At high intensities, solitons form, which have ( i ) a spectrum with discontinuous derivative, and ( k ) non-exponential decay in time. The units are arbitrary, and the intensity profiles in ( d , e , i ) and ( k ) are on logarithmic scales.
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second mitochondria-derived activator of caspases  (SMAC Corp)
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a – e Conventional solitons can form in the presence of quadratic dispersion. a Equivalence of the dispersion operator in the time and frequency domains. b The dispersion (red) and (inverse) group velocity (dashed green) vary smoothly with frequency ω . The dot dashed vertical line marks the soliton central frequency ω 0 . c At low intensities I , dispersion stretches pulses in time. Solitons can form at high intensities with smooth ( d ) spectral and ( e ) temporal profiles that decay exponentially. f Corresponding dispersion operator for Hilbert-NLS solitons. g The dispersion relation Eq. has a discontinuous (disc.) <t>derivative</t> (red) and associated discontinuous (inverse) group velocity (dashed green) as a function of frequency; h At low intensities, the dispersion causes input pulses to split in two. At high intensities, solitons form, which have ( i ) a spectrum with discontinuous derivative, and ( k ) non-exponential decay in time. The units are arbitrary, and the intensity profiles in ( d , e , i ) and ( k ) are on logarithmic scales.
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a – e Conventional solitons can form in the presence of quadratic dispersion. a Equivalence of the dispersion operator in the time and frequency domains. b The dispersion (red) and (inverse) group velocity (dashed green) vary smoothly with frequency ω . The dot dashed vertical line marks the soliton central frequency ω 0 . c At low intensities I , dispersion stretches pulses in time. Solitons can form at high intensities with smooth ( d ) spectral and ( e ) temporal profiles that decay exponentially. f Corresponding dispersion operator for Hilbert-NLS solitons. g The dispersion relation Eq. has a discontinuous (disc.) derivative (red) and associated discontinuous (inverse) group velocity (dashed green) as a function of frequency; h At low intensities, the dispersion causes input pulses to split in two. At high intensities, solitons form, which have ( i ) a spectrum with discontinuous derivative, and ( k ) non-exponential decay in time. The units are arbitrary, and the intensity profiles in ( d , e , i ) and ( k ) are on logarithmic scales.

Journal: Nature Communications

Article Title: Nonlinear wave propagation governed by a fractional derivative

doi: 10.1038/s41467-025-60625-4

Figure Lengend Snippet: a – e Conventional solitons can form in the presence of quadratic dispersion. a Equivalence of the dispersion operator in the time and frequency domains. b The dispersion (red) and (inverse) group velocity (dashed green) vary smoothly with frequency ω . The dot dashed vertical line marks the soliton central frequency ω 0 . c At low intensities I , dispersion stretches pulses in time. Solitons can form at high intensities with smooth ( d ) spectral and ( e ) temporal profiles that decay exponentially. f Corresponding dispersion operator for Hilbert-NLS solitons. g The dispersion relation Eq. has a discontinuous (disc.) derivative (red) and associated discontinuous (inverse) group velocity (dashed green) as a function of frequency; h At low intensities, the dispersion causes input pulses to split in two. At high intensities, solitons form, which have ( i ) a spectrum with discontinuous derivative, and ( k ) non-exponential decay in time. The units are arbitrary, and the intensity profiles in ( d , e , i ) and ( k ) are on logarithmic scales.

Article Snippet: To understand how a fractional Laplacian is represented in our optical experiment, we first consider the conventional case of the nonlinear Schrödinger equation, in which a second derivative operator (∣ β 2 ∣/2)(∂ 2 /∂ t 2 ), where β 2 is the dispersion coefficient, in the time domain, corresponds to a parabolic dispersion relation in frequency.

Techniques: Dispersion