Generalizing the interacting dilatonic ghost condensate as a dark energy model

In this article, we study the cosmic evolution of a generalized dilatonic ghost condensate field as a dark energy candidate, formulated from a Lagrangian density with two dominant kinetic terms; one linear and one of arbitrary integer $n>2$ in combination with an exponential potential, which interacts with dark matter through a source term. We analyzed three scenarios: the non-interacting situation $Q=0$ and two different interaction models, $Q\propto\rho_m\dot{\phi}$ and $Q\propto \rho_m H$ to describe the evolution of the present universe. For each interaction $Q$, we perform a detailed phase-space analysis to obtain stability conditions and identify critical points. In all situations, the system reproduces the standard cosmological dynamics and evolves toward late-time dark energy-dominated attractors, with quintessence or phantom features depending on the sign of the coupling parameter $\alpha$ associated with the standard kinetic term. Furthermore, a joint likelihood analysis with Cosmic Chronometers, PantheonPlus, and DESI observations is performed for two values of power $n$ ($n=3$ and $n=5$) to determine marginalized parameter constraints at the confidence levels of 68$\%$ and 95$\%$ for the different $Q-$models. For the interaction term $Q\propto \dot{\phi}\rho_m$, we find that the direction of the flow of energy depends on the sign of the coupling parameter $\alpha$ associated with the standard kinetic term. However, for the interaction $Q\propto H\,\rho_m$, the direction of the energy flow is independent of the sign of the coupling parameter $\alpha$ and always remains negative, corresponding to an energy transfer from dark matter to dark energy.