Integrable fractional modified Korteweg–deVries, sine-Gordon, and sinh-Gordon equations

The inverse scattering transform allows explicit construction of solutions to many physically significant nonlinear wave equations. Notably, this method can be extended fractional evolution equations characterized by anomalous dispersion using completeness suitable eigenfunctions the associated linear problem. In diffusion, mean squared displacement is proportional $t^{\alpha}$, $\alpha>0$, while in dispersion, speed localized waves $A^{\alpha}$, where $A$ amplitude wave. Fractional extensions modified Korteweg-deVries (mKdV), sine-Gordon (sineG) and sinh-Gordon (sinhG) hierarchies are obtained. Using symmetries present problem, these connected with a scalar family which mKdV (fmKdV), sineG (fsineG), sinhG (fsinhG) special cases. Completeness problem obtained and, from this, equation terms spectral expansion. particular, fmKdV, fsineG, fsinhG explicitly written. One-soliton derived for fmKdV fsineG solitons shown exhibit dispersion.