Related papers: Generalization of nonlocally related partial diffe…
Nonlocally related partial differential equation (PDE) systems are useful in the analysis of a given PDE system. It is known that each local conservation law of a given PDE system systematically yields a nonlocally related system. In this…
For partial differential equations (PDEs) that have $n\geq2$ independent variables and a symmetry algebra of dimension at least $n-1$, an explicit algorithmic method is presented for finding all symmetry-invariant conservation laws that…
For a system of partial differential equations (PDEs) $F = 0$ admitting a local (point, contact, or higher) symmetry $X$ with the characteristic $\varphi$, invariant solutions satisfy the reduced system $F = \varphi = 0$. We propose a…
Some connections between classical and nonclassical symmetries of a partial differential equation (PDE) are given in terms of determining equations of the two symmetries. These connections provide additional information for determining…
In this paper, we propose and study several inverse problems of determining unknown parameters in nonlocal nonlinear coupled PDE systems, including the potentials, nonlinear interaction functions and time-fractional orders. In these coupled…
Infinitesimal symmetries of a partial differential equation (PDE) can be defined algebraically as the solutions of the linearization (Frechet derivative) equation holding on the space of solutions to the PDE, and they are well-known to…
A simple characterization of the action of symmetries on conservation laws of partial differential equations is studied by using the general method of conservation law multipliers. This action is used to define symmetry-invariant and…
In the article we discuss the notion of the generalized invariant manifold introduced in our previous study. In the literature the method of the differential constraints is well known as a tool for constructing particular solutions for the…
Machine learned partial differential equation (PDE) solvers trade the reliability of standard numerical methods for potential gains in accuracy and/or speed. The only way for a solver to guarantee that it outputs the exact solution is to…
A general method using multipliers for finding the conserved integrals for any system of partial differential equations (PDEs) is reviewed and further developed in several ways. Multipliers are expressions whose (summed) product with a PDE…
We introduce a simple, rigorous, and unified framework for solving nonlinear partial differential equations (PDEs), and for solving inverse problems (IPs) involving the identification of parameters in PDEs, using the framework of Gaussian…
For a system of partial differential equations admitting point, contact, or higher symmetries, the framework of invariant reduction systematically computes how invariant geometric structures, such as conservation laws, presymplectic…
In this paper, symmetry analysis is extended to study nonlocal differential equations, in particular two integrable nonlocal equations, the nonlocal nonlinear Schr\"odinger equation and the nonlocal modified Korteweg--de Vries equation. Lie…
In this paper we consider generalization of procedure of construction of potential systems for systems of partial differential equations with multidimensional spaces of conservation laws. More precisely, for construction of potential…
We represent an integration algorithm combining the characteristics method and Hopf-Cole transformation. This algorithm allows one to partially integrate a large class of multidimensional systems of nonlinear Partial Differential Equations…
In this paper, we propose and study several inverse problems of identifying/determining unknown coefficients for a class of coupled PDE systems by measuring the average flux data on part of the underlying boundary. In these coupled systems,…
State-dependent parameter identification, where unknown model parameters depend on one or more state variables in partial differential equations (PDEs) or coupled PDE systems, is fundamental to a wide range of problems in physics,…
New nonlocal symmetries and conservation laws are derived for Maxwell's equations using a covariant system of joint vector potentials for the electromagnetic tensor field and its dual. A key property of this system, as well as of this class…
Nonlinear self-adjointness method for constructing conservation laws of partial differential equations (PDEs) is further studied. We show that any adjoint symmetry of PDEs is a differential substitution of nonlinear self-adjointness and…
We discuss nonlocal symmetries and nonlocal conservation laws that follow from the systematic potentialisation of evolution equations. Those are the Lie point symmetries of the auxiliary systems, also known as potential symmetries. We…