Weak solutions of unconditionally stable second-order difference schemes for nonlinear sine-Gordon systems

This paper presents the existence and uniqueness of the weak solution for the nonlinear system of sine-Gordon equations which describes DNA dynamics. An unconditionally stable second order difference scheme generated by the unbounded operator A2 corresponding to the system of sine-Gordon equations is considered. Weak solutions are a more general type of solution to the system of sine-Gordon equations than classical solutions and are important in the case of low regularity conditions. The weak solvability is studied in the space of distributions using variational methods. A very efficient numerical method that combines the finite difference method and the fixed point theory is used to perform numerical experiments to verify theoretical statements.