1. The problem is to understand and formulate the discrete lattice model described, which represents a nonlinear field theory on a lattice with back-reaction effects. 2. The model variables are: - $\phi_i(t)$: local field amplitude ("wiggle") at cell $i$. - $\pi_i(t)$: conjugate momentum at cell $i$. 3. The Hamiltonian (total energy) is given by: $$ H = \sum_i \left[ \frac{1}{2} \pi_i^2 + V(\phi_i) \right] + \frac{1}{2} \sum_{\langle i,j \rangle} K_{ij} (\phi_i - \phi_j)^2 $$ where the potential is nonlinear: $$ V(\phi) = \frac{1}{2} m^2 \phi^2 + \frac{\lambda}{4} \phi^4 $$ 4. The coupling $K_{ij}$ depends on local energy density (back-reaction): $$ K_{ij} = K_0 \left(1 + \alpha \frac{\rho_i + \rho_j}{2 E_P} \right) $$ where $\rho_i$ is the local energy density at cell $i$: $$ \rho_i = \frac{1}{2} \pi_i^2 + V(\phi_i) + \frac{1}{4} \sum_{j \in \text{nbrs}(i)} K_{ij} (\phi_i - \phi_j)^2 $$ 5. The equations of motion from Hamilton's equations are: $$ \dot{\phi}_i = \pi_i $$ $$ \dot{\pi}_i = - \frac{\partial V}{\partial \phi_i} + \sum_{j \in \text{nbrs}(i)} K_{ij} (\phi_j - \phi_i) - \frac{1}{2} \sum_{j \in \text{nbrs}(i)} (\phi_i - \phi_j) \frac{\partial K_{ij}}{\partial \phi_i} $$ The last term accounts for the dependence of $K_{ij}$ on $\phi_i$ via $\rho_i$ (nonlinear back-reaction). 6. For weak back-reaction, the last term can be neglected as a first approximation. 7. The complex field variant uses $\psi_i$ with Hamiltonian: $$ H = \sum_i \left[ |\pi_i|^2 + V(|\psi_i|) \right] + \sum_{\langle i,j \rangle} K_{ij} |\psi_i - \psi_j|^2 $$ where the phase of $\psi_i$ encodes charge and antimatter. This model describes a nonlinear lattice field theory with local potentials, elastic couplings, and back-reaction effects that modify coupling constants based on local energy density, supporting soliton-like excitations and complex phase dynamics.