Related papers: Generalized invariant manifolds for integrable equ…
The paper develops the method for construction of the families of particular solutions to the nonlinear Partial Differential Equations (PDE) without relation to the complete integrability. Method is based on the specific link between…
We develop a new integration technique allowing one to construct a rich manifold of particular solutions to multidimensional generalizations of classical $C$- and $S$-integrable Partial Differential Equations (PDEs). Generalizations of…
We introduce a constructive method that provides the local solution of general implicit systems in arbitrary dimension via Hamiltonian type equations. A variant of this approach constructs parametrizations of the manifold, extending the…
Physics-informed neural networks have emerged as a prominent new method for solving differential equations. While conceptually straightforward, they often suffer training difficulties that lead to relatively large discretization errors or…
Random invariant manifolds are geometric objects useful for understanding complex dynamics under stochastic influences. Under a nonuniform hyperbolicity or a nonuniform exponential dichotomy condition, the existence of random pseudo-stable…
A construction of differential constraints compatible with partial differential equations is considered. Certain linear determining equations with parameters are used to find such differential constraints. They generalize the classical…
We consider an ordinary nonlinear differential equation with generalized coefficients as an equation in differentials in algebra of new generalized functions. Then the solution of such equation will be a new generalized function. In the…
Transformations of differential equations to other equivalent equations play a central role in many routines for solving intricate equations. A class of differential equations that are particularly amenable to solution techniques based on…
The paper develops the method for construction of families of particular solutions to some classes of nonlinear Partial Differential Equations (PDE). Method is based on the specific link between algebraic matrix equations and PDE.…
Generalized equations are problems emerging in contexts of modern variational analysis as an adequate formalism to treat such issues as constraint systems, optimality and equilibrium conditions, variational inequalities, differential…
We develop a general framework for numerically solving differential equations while preserving invariants. As in standard projection methods, we project an arbitrary base integrator onto an invariant-preserving manifold, however, our method…
One of the very few mathematically rigorous nonlinear model reduction methods is the restriction of a dynamical system to a low-dimensional, sufficiently smooth, attracting invariant manifold. Such manifolds are usually found using local…
The article discusses a new method for constructing algebro-geometric solutions of nonlinear integrable lattices, based on the concept of a generalized invariant manifold (GIM). In contrast to the finite-gap integration method, instead of…
In this paper we construct nonlinear partial differential equations in more than 3 independent variables, possessing a manifold of analytic solutions with high, but not full, dimensionality. For this reason we call them ``partially…
Based on the matrix expression of general nonlinear numerical analogues presented by the present author, this paper proposes a novel philosophy of nonlinear computation and analysis. The nonlinear problems are considered an ill-posed linear…
This paper introduces, up to the author's knowledge, for the first time the generalized initial value problem. In this problem, given an ordinary differential equation defined in some set, the initial conditions are mapped to a subset of…
We construct manifold structures on various sets of solutions of the general relativistic initial data sets.
In numerous robotics and mechanical engineering applications, among others, data is often constrained on smooth manifolds due to the presence of rotational degrees of freedom. Common datadriven and learning-based methods such as neural…
In this letter, we introduce a new generalized linearizing transformation (GLT) for second order nonlinear ordinary differential equations (SNODEs). The well known invertible point (IPT) and non-point transformations (NPT) can be derived as…
We extend the construction of so-called encapsulated global summation-by-parts operators to the general case of a mesh which is not boundary conforming. Owing to this development, energy stable discretizations of nonlinear and variable…