  The intrinsic nonlinearity is the most remarkable characteristic of the
Bose-Einstein condensates (BECs) systems. Many studies have been done on atomic
BECs with time- and space- modulated nonlinearities, while there is few work
considering the atomic-molecular BECs with space-modulated nonlinearities.
Here, we obtain two kinds of Jacobi elliptic solutions and a family of rational
solutions of the atomic-molecular BECs with trapping potential and
space-modulated nonlinearity and consider the effect of three-body interaction
on the localized matter wave solutions. The topological properties of the
localized nonlinear matter wave for no coupling are analysed: the parity of
nonlinear matter wave functions depends only on the principal quantum number
$n$, and the numbers of the density packets for each quantum state depend on
both the principal quantum number $n$ and the secondary quantum number $l$.
When the coupling is not zero,the localized nonlinear matter waves given by the
rational function, their topological properties are independent of the
principal quantum number $n$, only depend on the secondary quantum number $l$.
The Raman detuning and the chemical potential can change the number and the
shape of the density packets. The stability of the Jacobi elliptic solutions
depends on the principal quantum number $n$, while the stability of the
rational solutions depends on the chemical potential and Raman detuning.
