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This paper deals with radial and non-radial solutions for local and non-local Liouville type equations. At first non- degenerate and degenerate mean field equations are studied and radially symmetric solutions to the Dirichlet problem for them are written into explicit form. The Cauchy boundary value problem for nonlinear Laplace equation with several exponential nonlinear- ities is considered and C2 smooth monotonically decreasing radial solution u(r) is found. Moreover, u(r) has logarithmic growth at ∞. Our results are applied to the differential geometry, more precisely, minimal non-superconformal degenerate two dimensional surfaces are constructed in R4 and their Gaussian, respectively normal curvature are written into explicit form. At the end of the paper several examples of local Liouville type PDE with radial coefficients which do not have radial solutions are proposed. [ABSTRACT FROM AUTHOR]

We discuss the Simple Equations Method (SEsM) for obtaining exact solutions of nonlinear partial differential equations. We show that a particular case of SEsM can be used in order to reproduce the methodology of the Inverse Scattering Transform Method for the case of the Burgers equation and Korteweg - de Vries equation. This particular case is connected to use of specific case of Step. 2 of SEsM: reprentation of the solution of the solved nonlinear partial differential equation as expansion as power series containing powers of a "small" parameter є, solving the differential equations arrising from this representation by means of Fourier series and transition from the obtained solution for small values of є to solution for arbitrary finite values of є. [ABSTRACT FROM AUTHOR]

In this paper, we conduct a comparative study between the homotopy perturbation method (HPM) and Adomian's decomposition method (ADM) for analytic treatment of nonlinear two-dimensional Volterra-Fredholm fuzzy integral equations (2D-VFFIE) and we show that the HPM with a specific convex homotopy is equivalent to the ADM for these type of equations. [ABSTRACT FROM AUTHOR]

In this paper a finite difference scheme for the one-dimensional double dispersion equation is constructed and studied. The scheme is based on the representation of this equation as a generalized Hamiltonian system. After applying the partitioned Runge-Kutta method with 3-stage Lobatto IIIA and IIIB coefficients, we get a discrete finite difference scheme with approximation error O(h2 + τ4). This scheme is symplectic, i.e. it preserves the symplectic structure of the solution on the discrete level. Numerical experiments are provided for quadratic nonlinearity. Two problems are considered: propagation of a single solitary wave and interaction of two waves traveling toward each other. The numerical results show the convergence of the discrete solution to the exact one with O(h2 + τ4) error. [ABSTRACT FROM AUTHOR]

Lipschitz stability for nonlinear differential equations with non-instantaneous impulses and supremum of the unknown function over a past time interval is studied. The impulses start abruptly at some points and their action continue on given finite intervals. Some sufficient conditions for Lipschitz stability are obtained. Modified Razumikhin method with piecewise continuous functions and comparison results with scalar non-instantaneous equations are applied. [ABSTRACT FROM AUTHOR]

In this paper, we propose the solution of partial Volterra fuzzy integro-differential equation (PVFIDE) with convolution type kernel using two-dimensional fuzzy Sumudu transform (2D-FST) method under Hukuhara differentiability. It is shown that 2D-FST is a simple and reliable approach for solving such equations analytically. Finally, the method is illustrated with example to show the ability of the proposed method. [ABSTRACT FROM AUTHOR]

We discuss the Simple Equations Method (SEsM) for obtaining exact solutions of nonlinear partial differential equations. We show that the much used Homogeneous Balance Method, Extended Homogeneous Balance method and Auxiliary Equation Method are particular cases of SEsM. [ABSTRACT FROM AUTHOR]

In this paper CNN modeling of tsunami waves is presented. Two models are studied: two-component Camassa-Holm type equation is studied and generalized KdV equation. For these cases CNN models are constructed and traveling wave solutions are obtained theoretically and via simulations. New type of traveling wave solutions are introduced — peak type, called peakon. Discussion and example of tsunami waves are provided at the end of the paper. [ABSTRACT FROM PUBLISHER]