In this article, we address the following question: Which hyperbolic or elliptic PDEs admit functional separable solutions. We shall focus on the study of a sinh-Gordon type equation. We construct solutions to this equation via the method of functional separation. We prove that these are the only families that have the property of functional separation and so we obtain a classification. To this end, we construct new families of solutions for the hyperbolic and elliptic versions of both sine and sinh-Gordon equations in a unified way.

The sine-Gordon (SG) partial differential equation (PDE) with an arbitrary perturbation is initially considered. Using the method of Kuzmak–Luke, we investigate the conditions, which the perturbation must satisfy, for a breather solution to be a valid leading-order asymptotic approximation to the perturbed problem. We analyse the cases of both stationary and moving breathers. As examples, we consider perturbing terms which include typical linear damping, periodic sinusoidal driving, and dispersion. The motivation for this study is that the mathematical modelling of physical systems often leads to the discrete SG system of ordinary differential equations, which are then approximated in the long wavelength limit by the continuous SG PDE. Such limits typically produce fourth-order spatial derivatives as correction terms. The new results show that the stationary breather solution is a consistent solution of both the quasi-continuum SG equation and the forced/damped SG system. However, the moving breather is only a consistent solution of the quasi-continuum SG equation and not the damped SG system.

The paper presents theoretical and numerical results on the identifiability, i.e. theunique identification for the one-dimensional sine-Gordon equation. The identifiabilityfor nonlinear sine-Gordon equation remains an open question. In this paper we establishthe identifiability for a linearized sine-Gordon problem. Our method consists of a carefulanalysis of the Laplace and Fourier transforms of the observation of the system, conductedat a single point. Numerical results based on the best fit to data method confirm that theidentification is unique for a wide choice of initial approximations for the sought testparameters. Numerical results compare the identification for the nonlinear and thelinearized problems.

In this paper the numerical solution of the two-dimensional sine-Gordon equation is studied. Split local artificial boundary conditions are obtained by the operator splitting method. Then the original problem is reduced to an initial boundary value problem on a bounded computational domain, which can be solved by the finite difference method. Several numerical examples are provided to demonstrate the effectiveness and accuracy of the proposed method, and some interesting propagation and collision behaviors of the solitary wave solutions are observed.

In this article we apply the optimal andthe robust control theory to the sine-Gordon equation. In our casethe control is given by the boundary conditions and we work in a finitetime horizon. We present at the beginning the optimal control problemand we derive a necessary condition of optimality and we continue byformulating a robust control problem for which existence and uniquenessof solutions are derived.