Hierarchy Of Partial Differential Equations Research Articles

Isospectral transformations in relativistic and nonrelativistic mechanics

The modified Korteweg de-Vries hierarchy of partial differential equations generating transformations of the one-dimensional Dirac equation, is shown to reduce in the limitc→∞ to the Korteweg de-Vries hierarchy, generating isospectral transformations of the Schrodinger equation. The former hierarchy reduces into relativistic and the latter into nonrelativistic isoperiodic transformation in the limitħ→0.

The homogeneous realization of the basic representation of A inf1 sup(1) and the toda lattice

It is well-known that the principal realization of the basic module L(Λ0) over Ainf1sup(1)gives rise to the KdV hierarchy of partial differential equations. Here we use the homogeneous realization of the same module to construct a hierarchy of differential-difference equations, the first member of which turns out to be the equation for the Toda lattice.

A simple approach to the Hamiltonian structure of soliton equations

A simple method for studying the Hamiltonian structure of the soliton equations is applied to a new hierarchy of partial differential equations related to a Zhakarov-Shabat-like spectral problem. These soliton equations are shown to be integrable Hamiltonian systems and can be described as motions on a symplectic Kähler manifold.

Perturbation Expansions on Perturbed Domains

Expansions in powers of a perturbation parameter are considered for partial differential equations to be solved on a domain that is also perturbed. Two kinds of expansions—Lagrange-like and Euler-like—are developed and shown to be equivalent to one another under a certain transformation of dependent variables. This equivalence is used to justify the hierarchy of partial differential equations produced by the Euler-like expansion. In addition to partial differential equations, integrals extended over a perturbed domain are also considered. Such integrals suffer perturbations due to the domain perturbation. A formula for these perturbations is given that involves the domain-mapping function only at the boundary of the unperturbed domain. This contrasts with the usual change-of-domain formula, which involves the Jacobian of the domain-mapping function throughout the unperturbed domain. The formula in question is compatible with Euler-like expansions (and the usual change-of-domain formula is compatible with Lagrange-like expansions), and is therefore appropriate in problems involving both partial differential equations and integrals on perturbed domains, for which the Euler-like viewpoint is preferable. Application is made to finding the gravitational potential of a perturbed mass, and related applications in astrophysical problems are indicated.

Tensorielle Kugelflächenfunktionsentwicklung des Stoßintegrals für ein schwach ionisiertes Plasma II

AbstractA spherical harmonic expansion of the deduced approximation [1] of the collision integral is obtained by expanding the electron distribution in spherical harmonics of the electron velocity. General spherical harmonic tensors are used. From the expansion a hierarchy of partial differential equations for the expansion tensors follows. This hierarchy is equivalent to the Boltzmann‐equation. In this paper the calculations are performed up to linear terms of the approximation.