  In this paper, we delve into the study of the generalized KP equation, which
incorporates double-power nonlinearities. Our investigation covers various
aspects, including the existence of solitary waves, their nonlinear stability,
and instability. Notably, we address a broader class of nonlinearities
represented by $\mu_1|u|^{p_1-1}u+\mu_2|u|^{p_2-1}u$, with $p_1>p_2$,
encompassing cases where $\mu_1>0$ and $\mu_1<0<\mu_2$. One of the distinct
features of our work is the absence of scaling, which introduces several
challenges in establishing the existence of ground states. To overcome these
challenges, we employ two different minimization problems, offering novel
approaches to address this issue. Furthermore, our study includes a nuanced
analysis to ascertain the stability of these ground states. Intriguingly, we
extend our stability analysis to encompass cases where the convexity of the
Lyapunov function is not guaranteed. This expansion of stability criteria
represents a significant contribution to the field. Moving beyond the analysis
of solitary waves, we shift our focus to the associated Cauchy problem. Here,
we derive criteria that determine whether solutions exhibit finite-time blow-up
or remain uniformly bounded within the energy space. Remarkably, our study
unveils a notable gap in the existing literature, characterized by the absence
of both theoretical evidence of blow-up and uniform boundedness. To explore
this intriguing scenario, we employ the integrating factor method, providing a
numerical investigation of solution behavior. This method distinguishes itself
by offering spectral-order accuracy in space and fourth-order accuracy in time.
Lastly, we rigorously establish the strong instability of the ground states,
adding another layer of understanding to the complex dynamics inherent in the
generalized KP equation.
