  Motivated by the observation of inhomogeneous patterns in some high-T$_c$
cuprate compounds, several variational Gutzwiller-projected wave-functions with
built-in charge and bond order parameters are proposed for the extended $t-J-V$
model on the square lattice at low doping. First, following a recent
Gutzwiller-projected mean-field approach by one of us (Phys. Rev. B. {\bf 72},
060508(R) (2005)), we investigate, as a function of doping and Coulomb
repulsion, the stability of the staggered flux phase with respect to small
spontaneous modulations of squared unit cells ranging from $2\times 2$ to
$\sqrt{32}\times\sqrt{32}$. It is found that a $4\times 4$ bond-order (BO)
modulation appears spontaneously on top of the staggered flux pattern for hole
doping around 1/8. A related wave-function is then constructed and optimized
accurately and its properties studied extensively using an approximation-free
variational Monte Carlo scheme. Finally, the competition of the BO-modulated
staggered flux wave-function w.r.t. the d-wave RVB wave-function or the
commensurate flux state is investigated. It is found that a short range Coulomb
repulsion penalizes the d-wave superconductor and that a moderate Coulomb
repulsion brings them very close in energy. Our results are discussed in
connection to the STM observations in the under-doped regime of some cuprates.

Strongly Correlated Electrons;Superconductivity