| Abstract: |
| We study a coupled gKdV-NLS system $u_t + \alpha u^p u_x + \beta u_{xxx} & = \gamma (|\psi|^2)_x$ and $i\psi_t + \kappa \psi_{xx} = \sigma u \psi$ with a general nonlinearity power $p>0$, which has been introduced in the literature to model energy transport in an anharmonic crystal material [1,2]. There is a strong interest in obtaining exact solutions describing frequency-modulated solitary waves $u=U(x-ct)$, $\psi=e^{i\w t}\Psi(x-ct)$, where $c$ is the wave speed, and $\w$ is the modulation frequency. For the KdV case $p=1$, some solutions have been found in [1], while for the mKdV case $p=2$, no exact solutions were found [2]. Nothing has been done for higher nonlinearities $p\geq 3$.
In the present work, we derive 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 [3];
(ii) apply a hodograph transformation which leads to triangular (decoupled) 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 coefficients in the ansatz under those conditions.
The resulting solitary waves exhibit a wide range of features: bright and dark peaks; single peaked and multi-peaked; zero and non-zero backgrounds.
[1] Physica D 346 (2017), 20-27.
[2] Physics Letters A 382 (2018), 837--845.
[3] Commun. Nonlin. Sci. Numer. Simul. 91 (2020), 105349. |
|