In this paper, based on sinh-cosh method and sinh-Gordon expansion method,families of solutions of (2+1)-dimensional breaking soliton equation are obtained.These solutions include Jacobi elliptic function solution, soliton solution,trigonometric function solution.

We prove that a directed last passage percolation model with discontinuous macroscopic (non-random) inhomogeneities has a continuum limit that corresponds to solving a Hamilton-Jacobi equation in the viscosity sense. This Hamilton-Jacobi equation is closely related to the conservation law for the hydrodynamic limit of the totally asymmetric simple exclusion process. We also prove convergence of...

in this paper, based on sinh-cosh method and sinh-gordon expansion method,families of solutions of (2+1)-dimensional breaking soliton equation are obtained.these solutions include jacobi elliptic function solution, soliton solution,trigonometric function solution.

The nontrivial transformation of the phase space path integral measure under certain discretized analogues of canonical transformations is computed. This Jacobian is used to derive a quantum analogue of the Hamilton-Jacobi equation for the generating function of a canonical transformation that maps any quantum system to a system with a vanishing Hamiltonian. A formal perturbative solution of th...

1 Linear Equation Systems in the Numerical Solution of PDE’s 3 1.1 Examples of PDE’s . . . . . . . . . . . . . . . . . . . . . . . . 3 1.2 Weak Formulation of Poisson’s Equation . . . . . . . . . . . . 6 1.3 Finite-Difference-Discretization of Poisson’s Equation . . . . . 7 1.4 FD Discretization for Convection-Diffusion . . . . . . . . . . 8 1.5 Irreducible and Diagonal Dominant Matrices . . . ...

The Jacobi equation in pseudo-Riemannian geometry determines the linearized geodesic flow. The linearization ignores the relative velocity of the geodesics. The generalized Jacobi equation takes the relative velocity into account; that is, when the geodesics are neighboring but their relative velocity is arbitrary the corresponding geodesic deviation equation is the generalized Jacobi equation....

We establish well-posedness of a class of first order Hamilton-Jacobi equation in geodesic metric spaces. The result is then applied to solve a Hamilton-Jacobi equation in the Wasserstein space of probability measures, which arises from the variational formulation of a compressible Euler equation.

The geometric structure of the Hamilton-Jacobi equation is analyzed by using symplectic geometry. Generating function of symplectic transform plays an important role. It will be shown that the Hamilton-Jacobi equation possesses important geometric properties such as existence condition and maximality of the stabilizing solution, which are well-known in the Riccati equation.

In general, the value function associated with an exit time problem is a discontin-uous function. We prove that the lower (upper) semicontinuous envelope of the value function is a supersolution (subsolution) of the Hamilton-Jacobi equation involving the proximal subdiierentials (superdiierentials) with subdiierential type (superdiierential type) mixed boundary condition. We also show that if t...