Toronto, December 2 - 5, 2022
This is a joint work with Fabio Pusateri.
We derived exact solutions for $p=1,2,3,4$, starting from the travelling wave ODE-system satisfied by $U$ and $\Psi$. The method is new: (i) obtain first integrals by use of multi-reduction symmetry theory ; (ii) apply a hodograph transformation which leads to a triangular system; (iii) introduce an ansatz for polynomial solutions of the base ODE; (iv) characterize conditions under which solutions yield solitary waves; (v) solve an algebraic system for the unknown coefficients under those conditions.
The resulting solitary waves exhibit a wide range of features: bright/dark peaks; single/multi-peaked; zero/non-zero backgrounds.
 L.A. Cisneros-Ake, J.F. Solano Pelaez, Bright and dark solitons in the unidirectional long wave limit for the energy transfer on anharmonic crystal lattices, Physica D 346 (2017), 20--27.
 L.A. Cisneros-Ake, H. Parra Prado, D.J. Lopez Villatoro, R. Carretero-Gonzalez, Multi-hump bright solitons in a Schrodinger–mKdV system, Physics Letters A 382 (2018), 837--845.
 S.C. Anco and M.L. Gandarias,
Symmetry multi-reduction method for partial differential equations with conservation laws,
Commun. Nonlin. Sci. Numer. Simul. 91 (2020), 105349.