We examine how symmetry and computer algebra can assist in solving the Cauchy problem for Pfaffian systems. We use recent results on integrating Frobenius integrable distributions via solvable symmetry structures to develop two techniques that when used in conjunction with symmetry determination software DIMSYM, allow us to solve the Cauchy problem for the special situation when there exists a one-dimensional Cauchy characteristic space. We also illustrate how our work can assist in extracting local solutions of a certain class of first and second order non-linear partial differential equations.

We prove the existence of energy equilibria of the di-block copolymer problem in the unit disk. They consist of concentrically layered micro-domains rich in one of the two monomer building units. We construct them by solving the proper singular limit of the free energy functional. The same limit also explains how under a dynamic law of the free energy, circular interfaces of non-equilibria may move to the origin and vanish, or collapse to each other, thereby reducing the number of layers.

We consider the two-dimensional Rayleigh–Taylor problem for the dynamics of the free interface Γ between two layers of immiscible viscous liquids. For a slow flow model (which corresponds to the case of a small relative jump of density) and under sufficiently wide assumptions on the geometry of Γ, we analyze the time dynamics of Γ. In particular, we prove that its increase in time t is bounded by an exponential function with exponent independent of Γ.

Dorsey et al. [8] have constructed formal solutions for the half-space Ginzburg–Landau model, when κ is small. Dorsey et al. deduce a formal expansion for the superheating field in powers of κ½ up to order 4. In this paper, we show how the formal construction gives a natural way for constructing a subsolution for the Ginzburg–Landau system. We improve the result obtained by Bolley & Helffer [2], and take a step in the proof of the Parr Formula [13], getting two terms in the lower bound for the superheating field as κ → 0.

An effective algorithmic method is presented for finding the local conservation laws for partial differential equations with any number of independent and dependent variables. The method does not require the use or existence of a variational principle and reduces the calculation of conservation laws to solving a system of linear determining equations similar to that for finding symmetries. An explicit construction formula is derived which yields a conservation law for each solution of the determining system. In the first of two papers (Part I), examples of nonlinear wave equations are used to exhibit the method. Classification results for conservation laws of these equations are obtained. In a second paper (Part II), a general treatment of the method is given.

This paper gives a general treatment and proof of the direct conservation law method presented in Part I (see Anco & Bluman [3]). In particular, the treatment here applies to finding the local conservation laws of any system of one or more partial differential equations expressed in a standard Cauchy-Kovalevskaya form. A summary of the general method and its effective computational implementation is also given.