In [P5] was proposed a new (vectorial) interpretation of the operator-splitting schemes for Navier-Stokes equations in terms of vorticity/stream function. The new kind of splitting leaves the system coupled for each fractional time step allowing the satisfaction of the two conditions on the stream function. Thus no artificial non-physical conditions on vorticity were needed and the scheme became fully implicit.
In [J76], [P46] the problems of splitting are addressed for the case when the diffusion is represented by a fourth-order operator. A new scheme for the so-called Swift-Hohenberg equation is developed which allows obtaining results for very long times. A similar idea is applied to Navier-Stokes Equations in the stream-function formulation in [P31], [J68] where the fourth order operator is split instead of introducing the auxiliary vorticity function.
The fully implicit coordinate splitting is being now further developed for the Navier-Stokes equation in primitive variables [P49]. Currently, the application of the splitting algorithm to thermocapillary and buoyancy driven flows is under way.
For generalized dispersive equations (Boussinesq, coupled Schrödinger, etc.) strongly implicit difference schemes are developed which faithfully represent the conservation laws holding for the original system. The long-time evolution of the solution is investigated [J67], [P35], [P39], [J65], [P40], [J63].
For dissipative models, implicit schemes representing the balance between the energy input and dissipation are created [P35], [J62], [J64].
In [J25] we proposed a complete orthonormal system in L2(-∞,∞) with the novel (at that time) property of having formula expressing a product of two members of the system into a series in the system. This was essential for creating Fourier-Galerkin techniques [J46], [J55] for numerical investigation of localized solutions of nonlinear equations, e.g., KdV, Burgers, HKS. The technique is successfully applied also to FKDV equation [J58] and to an integro-differential equation arising in interfacial hydrodynamics [J45]. The application to the 2D case is elaborated in [J72] for the cubic Klein-Gordon equation. Recently, the techniques is being developed further in application to the case of cubic nonlinearity ([J85]) and for localized solutions with damped oscillatory tails ([J52]).
This is a new approach to devising robust difference schemes and algorithms for numerical solution of inverse or ill-posed problems. The gist of MVI is the replacement of the ill-posed problem with the correct boundary value problem for the Euler-Lagrange equations giving the necessary conditions for minimization of the quadratic functional of the original equations. The first application of MVI is in [P14] where the homoclinic trajectory of Lorenz system is calculated. In a similar manner in [J60], [P20], [P26], [P36] the localized solutions of KSV equation are computed. The application of MVI to the classical inverse problems of analytical continuation is sketched in [J40]. A difference scheme for coefficient identification based on MVI is developed in [J41], [J78]. The boundary layer flow is treated as inverse problem of coefficient identification is treated in [P28], [J77]. MVI is applied also for identification of steady solutions of Navier-Stokes equations for high Reynolds numbers when these are unstable [P30], [J66].
New models of nonlinear wave phenomena are proposed for different physical systems where the conservation and balance laws are of primary importance leading to appearance of solitary-wave solutions (e.g., solitons). Here belong the works on localized waves as quasi particles: [P34], [P41], [J75], [P54]; the new models of thin elastic layers [P34], [P41], [J71]; further development of the Boussinesq Paradigm (balance between nonlinearity and dispersion) [J73], [P55], [J86];
The dynamics of solitons in conservative systems is investigated for very long times making use of the conservative difference schemes above developed [J63], [J65], [J67], [P29], [P35], [P39], [P40], [J73]. The physical relevance of the well known sech-shaped solitons is assessed and the inelasticity of their collisions thoroughly investigated. Other classes of solutions are found numerically that also qualify for solitons in the sense of preservation of shape and conservation of energy and momentum. These are self-similar solutions whose support increases and the amplitude decreases with time while the total mass and energy of the solitary wave remain constant.
The long-time evolution and interaction of localized waves in dissipative and non-integrable systems (Nonlinear Evolution Equations descendants of the Homsy-Kuramoto-Sivashinsky equation) are investigated in [P34], [P37], [J62], and soliton-like behavior of the solitary waves is unraveled in [J69]. With a proper justification this kind of "quasi-particles" can be called dissipative solitons. This research was taken up to a new frontier considering Generalized Nonlinear Wave Equations [J64], [P51] in lieu of the well studied Nonlinear Evolution Equations. It is shown that even in the intrinsically dissipative case the coherent structures can behave as quasi-particles. This extends the concept of balance to the case when the balance is between the energy production, energy dissipation and the nonlinearity.
Pattern formation in dissipative system with energy input is investigated in [J74], [J76], [P50], [J83], [J84], on the basis of Swift-Hohenberg's and Knobloch's (1+2)D models.
Dynamics of energy-conserving coupled maps lattice arising from an implicit difference approximation of the cubic Klein-Gordon equation (φ4 potential) is investigated numerically in [J70] and a chaotic regime is uncovered. The approach to statistical equilibrium depends on the energy density and the size of the system and the numerical approach developed allows one to assess the parametric regions. A new kind of intermittency is found in which the trajectory of the system switches stochastically between two manifolds.