Showing papers by "Wolfgang K. Schief published in 2014"

The classical Korteweg capillarity system: geometry and invariant transformations

Abstract: A class of invariant transformations is presented for the classical Korteweg capillarity system. The invariance is an extension of a kind originally introduced in an anisentropic gasdynamics context. In a particular instance, application of the invariant transformation leads to a deformed one-parameter class of Karman–Tsien-type capillarity laws associated with a deformation of an integrable nonlinear Schrodinger-type equation which incorporates a de Broglie–Bohm potential. The latter and another integrable case associated with the classical Boussinesq equation may be linked to the motion of curves in Euclidean and projective space so that both the invariant transformation and the Galilean invariance of the capillarity system may be interpreted in a geometric and soliton-theoretic manner. The work is set in the broader context of other connections of invariant transformations in gasdynamics with soliton theory.

Critical points, Lauricella functions and Whitham-type equations

Abstract: A large class of semi-Hamiltonian systems of hydrodynamic type is interpreted as the equations governing families of critical points of functions obeying the classical linear Darboux equations for conjugate nets.The distinguished role of the Euler-Poisson-Darboux equations and associated Lauricella-type functions is emphasised. In particular, it is shown that the classical g-phase Whitham equations for the KdV and NLS equations are obtained via a g-fold iterated Darboux-type transformation generated by appropriate Lauricella functions.

Integrable discretization of hodograph-type systems, hyperelliptic integrals and Whitham equations

Abstract: Based on the well-established theory of discrete conjugate nets in discrete differential geometry, we propose and examine discrete analogues of important objects and notions in the theory of semi-H...

On Weingarten Transformations of Hyperbolic Nets

Abstract: Weingarten transformations of smooth surfaces, which, by definition, preserve asymptotic lines have been studied extensively in classical differential geometry, and they also play an important role in connection with the modern geometric theory of integrable systems. Their natural discrete analogs have been investigated in great detail in the area of (integrable) discrete differential geometry and can be traced back at least to the early 1950s. Here, we propose a canonical analog of (discrete) Weingarten transformations for hyperbolic nets, that is, C 1 -surfaces which constitute hybrids of smooth and discrete surfaces “parameterized” in terms of asymptotic coordinates. We prove the existence of Weingarten pairs and analyze their geometric and algebraic properties.

On a Boussinesq Capillarity System: Hamiltonian Reductions and Associated Quartic Geometries

Abstract: A nonlinear Boussinesq-type capillarity model system is shown to admit exact reductions to coupled Hamiltonian subsystems associated with time-modulated quartic density distributions. In particular, in 3+1 dimensions, time-modulated “red blood cell” geometries for the density distribution are isolated.

On q-Gaussian Integrable Hamiltonian Reductions in Anisentropic Magneto-gasdynamics

Abstract: Integrable substructure in 2+1-dimensional anisentropic magneto-gasdynamics is investigated via a general elliptic vortex ansatz. The procedure involves introduction of a q-Gaussian density representation. Thermodynamically consistent relations are isolated associated with certain integrable Hamiltonian reductions.

Integrable structure in discrete shell membrane theory

Abstract: We present natural discrete analogues of two integrable classes of shell membranes. By construction, these discrete shell membranes are in equilibrium with respect to suitably chosen internal stresses and external forces. The integrability of the underlying equilibrium equations is proved by relating the geometry of the discrete shell membranes to discrete O surface theory. We establish connections with generalized barycentric coordinates and nine-point centres and identify a discrete version of the classical Gauss equation of surface theory.

Discrete line complexes and integrable evolution of minors

Abstract: Based on the classical Plucker correspondence, we present algebraic and geometric properties of discrete integrable line complexes in $CP^3$. Algebraically, these are encoded in a discrete integrable system which appears in various guises in the theory of continuous and discrete integrable systems. Geometrically, the existence of these integrable line complexes is shown to be guaranteed by Desargues' classical theorem of projective geometry. A remarkable characterisation in terms of correlations of $CP^3$ is also recorded.

Integrable discretization of hodograph-type systems, hyperelliptic integrals and Whitham equations

Abstract: Based on the well-established theory of discrete conjugate nets in discrete differential geometry, we propose and examine discrete analogues of important objects and notions in the theory of semi-Hamiltonian systems of hydrodynamic type. In particular, we present discrete counterparts of (generalised) hodograph equations, hyperelliptic integrals and associated cycles, characteristic speeds of Whitham type and (implicitly) the corresponding Whitham equations. By construction, the intimate relationship with integrable system theory is maintained in the discrete setting.