Algebraic properties of Gardner's deformations for integrable systems

An algebraic definition of Gardner's deformations for completely integrable bi-Hamiltonian evolutionary systems is formulated. The proposed approach extends the class of deformable equations and yields new integrable evolutionary and hyperbolic Liouville-type systems. An exactly solvable two-component extension of the Liouville equation is found.Comment: Proc. conf. "Nonlinear Physics: Theory and Experiment IV" (Gallipoli, 2006); Theor. Math. Phys. (2007) 151:3/152:1-2, 16p. (to appear

A modified Galerkin/finite element method for the numerical solution of the Serre-Green-Naghdi system

A new modified Galerkin / Finite Element Method is proposed for the numerical solution of the fully nonlinear shallow water wave equations. The new numerical method allows the use of low-order Lagrange finite element spaces, despite the fact that the system contains third order spatial partial derivatives for the depth averaged velocity of the fluid. After studying the efficacy and the conservation properties of the new numerical method, we proceed with the validation of the new numerical model and boundary conditions by comparing the numerical solutions with laboratory experiments and with available theoretical asymptotic results

Modeling Nonlinear Dispersive Water Waves

An expository review is given on various theories of modeling weakly to strongly nonlinear, dispersive, time-evolving, three-dimensional gravity-capillary waves on a layer of water. It is based on a new model that allows the nonlinear and dispersive effects to operate to the same full extent as in the Euler equations. Its relationships with some existing models are discussed. Various interesting phenomena will be illustrated with applications of these models and with an exposition on the salient features of nonlinear waves in wave-wave interactions and the related processes of transport of mass and energy

On the notion of conditional symmetry of differential equations

Symmetry properties of PDE's are considered within a systematic and unifying scheme: particular attention is devoted to the notion of conditional symmetry, leading to the distinction and a precise characterization of the notions of ``true'' and ``weak'' conditional symmetry. Their relationship with exact and partial symmetries is also discussed. An extensive use of ``symmetry-adapted'' variables is made; several clarifying examples, including the case of Boussinesq equation, are also provided.Comment: 18 page