Robert E. L. Turner

Broadening of interfacial solitary waves

Progressing interfacial gravity waves are considered for two fluids of differing densities confined in a channel of finite vertical extent and infinite horizontal extent. An integrodifferential equation for the unknown shape of the interface is derived. This equation is discretized and the resulting algebraic equations are solved using Newton’s method. It is found that, for a range of heights and densities of the two fluids, the system supports a branch of solitary waves. Progression along the branch produces a broadening of the wave. With increased broadening both the amplitude and the wave speed approach limiting values. The results are in good agreement with analytical studies and indicate the existence of internal surges.

Some variational principles for a nonlinear eigenvalue problem

where L and M are linear differential operators on a finite interval [u, 61 (cf. [I], p. 430). If one works in L,[a, b], in typical cases the problems are such that L is self-adjoint, and M is an operator of lower order, usually symmetric. This problem was examined by Shinbrot [2] who wrote it in the form Au = Au + h2Bu with h = p-l, A = L-l and B = -L-lM (ML-1 seems preferable if one wants to show equivalence with (1.1)). He then treated the problem Au = Au + kB,p with a! > 1, A = A* compact, and B, a bounded linear operator, Lipschitz continuous in h with respect to the operator-norm. Under conditions on the location of the spectrum of A and on the size of the norms 11 A 11 and 11 B, 11, he showed that the nonlinear eigenvalue problem has a sequence of eigenvalues converging to zero and a corresponding basis of eigenvectors. Here, with some conditions on L and M, we reduce (1.1) to a problem

The limiting configuration of interfacial gravity waves

Progressive gravity waves at the interface between two unbounded fluids are considered. The flow in each fluid is taken to be potential flow. The problem is converted into a set of integrodifferential equations, reduced to a set of algebraic equations by discretization, and solved by Newton’s method together with parameter variation. Meiron and Saffman’s [J. Fluid Mech. 129, 213 (1983)] calculations showing the existence of overhanging waves are confirmed. However, the present calculations do not support Saffman and Yuen’s [J. Fluid Mech. 123, 459 (1982)] conjecture that the waves are geometrically limited (i.e., that solutions exist until the interface intersects itself). It is proposed that along a solution branch starting with sinusoidal waves of small amplitude, one reaches solutions with vertical streamlines and then overhanging waves. Continuing on this branch, one returns to nonoverhanging waves and then back toward a wave with vertical streamlines. It is suggested that this succession of patterns ...

Matrices Hermitian for an Absolute Norm

Letv be a (standardized) absolute norm onC:n . A matrix Hin Cnn is called norm-Hermitian if the numerical range V(H) determined by vis real. Let be the set of all norm-Hermitians in Cnn .We determine an equivalence relation ∼ on {1,…,n) with the following property: Let H e Cnn Then if and only if His Hermitian and . Let is a subalgebra of Cnn and, for V(A) equals the Euclidean numerical range and hence is convex. Let be the group of isometries forv,and let Then is a normal subgroup of and where is a group of permutation matrices.

Long periodic internal waves

Long periodic waves propagating in a channel bounded above and below by horizontal walls are considered. The fluid consists of two layers of constant densities separated by a region in which the density varies continuously. The problem is solved numerically by a finite difference scheme coupled with boundary integral equation techniques. It is shown that there are long periodic waves characterized by a train of ripples in their troughs. The numerical results suggest the existence of a wave with one large crest flanked on either side by a small‐amplitude oscillatory wave extending to infinity, in anology with the ‘‘solitary wave with oscillatory tail,’’ known to exist for surface waves with small surface tension.

Diagonal norm hermitian matrices

Abstract If v is a norm on C n, let H(v) denote the set of all norm-Hermitians in C nn. Let S be a subset of the set of real diagonal matrices D. Then there exists a norm v such that S=H(v) (or S = H(v)∩D) if and only if S contains the identity and S is a subspace of D with a basis consisting of rational vectors. As a corollary, it is shown that, for a diagonable matrix h with distinct eigenvalues λ1,…, λr, r⩽n, there is a norm v such that h ∈ H(v), but hs∉H(v), for some integer s, if and only if λ2–λ1,…, λr–λ1 are linearly dependent over the rationals. It is also shown that the set of all norms v, for which H(v) consists of all real multiples of the identity, is an open, dense subset, in a natural metric, of the set of all norms.

Center manifolds in equations from hydrodynamics

Elliptic partial differential equations on tubular domains arise in hydrodynamics and in other applied settings. The center manifold approach to the analysis of small solutions has as one of its steps the study of a reduced equation of differential-integral type leading to a system of ordinary differential equations, generally of finite order. Here we give existence and regularity results for reduced equations which are adapted to the Hölder space setting and which are suitable for application to quasi-linear problems. Applications to flow problems are given.

Perturbation of ordinary differential operators

Abstract A perturbation theorem is proved which enables us to extend the results of J. Schwartz and others, to the effect that a wide class of boundary value problems for ordinary differential operators with operator valued coefficients generate unbounded spectral operators in the L2 space over a finite interval. Thus under mild restrictions on the boundary conditions allowed, an operator ( 1 i d dx ) n + C n−1 ( d dy ) n−1 + B n−2 ( d dx ) n−2 + ··· + B 0 in L2[0, 1], where B0, …, Bn−2 are arbitrary bounded operators and Cn−1 is any compact operator plus a bounded one with small norm, is shown to have a complete set of generalized eigenfunctions (φn). Moreover, the series expansions in the φn are unconditionally convergent.