The study of the separability of the Hamilton–Jacobi and the corresponding scalar field equations in a curved spacetime has a long history. Robertson [1] and Eisenhart [2] discussed general conditions for such a separability in spaces which admit a complete set of mutually orthogonal families of hypersurfaces. An important class of 4-dimensional separable spacetimes, including several type D me...

We show that for a Jacobi operator with coefficients whose (j + 1)’th moments are summable the j’th derivative of the scattering matrix is in the Wiener algebra of functions with summable Fourier coefficients. We use this result to improve the known dispersive estimates with integrable time decay for the time dependent Jacobi equation in the resonant case.

In this work, we propose a high resolution Alternating Evolution Discontinuous Galerkin (AEDG) method to solve Hamilton-Jacobi equations. The construction of the AEDG method is based on an alternating evolution system of the Hamilton-Jacobi equation, following the previous work [H. Liu, M. Pollack and H. Saran, SIAM J. Sci. Comput. 35(1), (2013) 122–149] on AE schemes for Hamilton-Jacobi equati...

‎In this paper‎, ‎we obtain exact solutions involving parameters of some nonlinear PDEs in mathmatical physics; namely the one-‎dimensional modified complex Ginzburg-Landau equation by using the $ (G'/G) $ expansion method‎, homogeneous balance method, extended F-expansion method‎. ‎By ‎using homogeneous balance principle and the extended F-expansion, more periodic wave solutions expressed by j...

Using a semi-discrete model that describes the heat transfer of a continuous casting process of steel, this paper is addressed to an optimal control problem of the continuous casting process in the secondary cooling zone with water spray control. The approach is based on the Hamilton–Jacobi–Bellman equation satisfied by the value function. It is shown that the value function is the viscosity so...

We show that an invariant surface allows to construct the Jacobi vector field along a geodesic and construct the formula for the normal component of the Jacobi field. If a geodesic is the transversal intersection of two invariant surfaces (such situation we have, for example, if the geodesic is hyperbolic), then we can construct a fundamental solution of the the Jacobi-Hill equation ü = −K(u)u....

We introduce a novel numerical method for a recently developed perspective Shape-from-Shading model. In order to discretise the corresponding partial differential equation (PDE), Prados et al. employed the dynamical programming principle yielding a Hamilton-Jacobi-Bellman equation. We reduce that model to its essential, namely to the underlying Hamilton-Jacobi equation. For this PDE, we propose...

By using solutions of an ordinary differential equation, an auxiliary equationmethod is described to seek exact solutions of variablecoefficient KdV-MKdV equation. As a result, more new exact nontravelling solutions, which include soliton solutions, combined soliton solutions, triangular periodic solutions, Jacobi elliptic function solutions, and combined Jacobi elliptic function solutions, for...