# Similarity: Generalizations, Applications and Open Problems

## Topic

**Conservation law methods using multipliers and symmetries**

*Stephen Anco*

Brock University

Brock University

Time: Monday 16:55-17:35

The first part of this talk will review the basic methods for finding conservation laws of PDEs. In the second part, a direct method will be described for obtaining conservation laws invariant under a given symmetry group of a PDE. Examples and applications to nonlinear evolution equations will be used to illustrate the method.

**Stability and Dynamics of Self-similarity in Evolution Equations**

*Andrew J. Bernoff*

Department of Mathematics

Harvey Mudd College

Department of Mathematics

Harvey Mudd College

Time: Monday 9-9:40

Similarity methods have been used to derive special solutions for a broad variety of physical problems in the past few decades. In this talk I will discuss a methodology for studying linear stability for self-similar blow-up and pinch-off. I will present three problems: a simple ODE model, the Sivashinsky equation which arises in solidification, and the pinch-off of a solid filament due to the action of surface diffusion. The goal is to show that self-similar phenomena can be studied using many of the now familiar ideas that have arisen in the study of dynamical systems. In particular, I will discuss rescaling methods, linearization and the role of symmetries in the context of self-similarity. I will demonstrate that the symmetries in the problem give rise to "anomalous" positive eigenvalues associated with the rescaling symmetries as opposed to instability, and show how this stability analysis can identify a unique stable (and observable) solution from a countable infinity of similarity solutions.

**Asymptotic dynamics of attractive-repulsive swarms**

**Andrew J. Bernoff**

Dept. of Mathematics, Harvey Mudd College, Claremont, CA 91711

Dept. of Mathematics, Harvey Mudd College, Claremont, CA 91711

*Andrew J. Leverentz (HMC 2008)*

Dept. of Mathematics, Harvey Mudd College, Claremont, CA 91711

Dept. of Mathematics, Harvey Mudd College, Claremont, CA 91711

*Chad M. Topaz*

Dept. of Mathematics and Computer Science, Macalester College, St. Paul, MN 55105

Dept. of Mathematics and Computer Science, Macalester College, St. Paul, MN 55105

Time: Tuesday 16:35-16:50

We classify and predict the asymptotic dynamics of a class of swarming models. The model consists of a conservation equation in one- dimension describing the movement of a population density field. The velocity is found by convolving the density with a kernel describing attractive-repulsive social interactions. The kernel's first moment and its limiting behavior at the origin determine whether the population asymptotically spreads, contracts, or reaches steady- state. For the spreading case, the dynamics approach those of the porous medium equation. The widening, compactly-supported population has edges that behave like traveling waves whose speed, density and slope we calculate. For the contracting case, the dynamics of the cumulative density approach those of Burgers' equation. We derive an analytical upper bound for the finite blow-up time after which the solution forms one or more $\delta$-functions.

**Blowup in multidimensional aggregation equations with mildly singular interaction kernels**

*Andrea Bertozzi*

Univ. of California Los Angeles

Univ. of California Los Angeles

Time: Wednesday 16:55-17:35

We consider the multidimensional aggregation equation $u_t -\nabla\cdot (u \nabla K*u) = 0$ in which the radially symmetric attractive interaction kernel has a mild singularity at the origin (Lipschitz or better). In the case of bounded initial data, finite time singularity has been proved for kernels with a Lipschitz point at the origin, whereas for C2 kernels there is no finite-time blowup. We prove, under mild monotonicity assumptions on the kernel $K$, that the Osgood condition for well-posedness of the ODE characteristics determines global in time well-posedness of the PDE with compactly supported bounded

nonnegative initial data. When the Osgood condition is violated, we present a new proof of finite time blowup that extends previous results, requiring radially symmetric data, to general bounded, compactly supported nonnegative initial data without symmetry. We

also present a new analysis of radially symmetric solutions under less strict monotonicity conditions. Finally we conclude with a discussion of similarity solutions for the case $K(x)=|x|$ and some open problems.

This is joint work with Jose Carrillo (Barcelona) and Thomas Laurent (UCLA).

**Hidden symmetries of the ideal MHD equilibrium equations**

*Oleg Bogoyavlenskij*

Department of Mathematics, Queenâ€™s University, Kingston, K7L 3N6 Canada

Department of Mathematics, Queenâ€™s University, Kingston, K7L 3N6 Canada

Time: Tuesday 11:45-12:25

Intrinsic symmetries of the ideal magnetohydrodynamics equilibrium equations are introduced for the divergence-free plasma flows. The intrinsic symmetries break the geometrical symmetries of the field-aligned plasma equilibria. The symmetries form infinite-dimensional abelian Lie groups and depend upon two arbitrary functions a(x) and b(x) that are constant on the magnetic field lines and on the plasma streamlines. Applying the new symmetries gives both the global non-symmetric MHD equilibria which model the astrophysical jets and the exact MHD equilibria with non-collinear vector fields B and V.

**Exact solution for nonlinear unsaturated flow with Dirichlet boundary conditions.**

**P. Broadbridge* and D. Triadis, Australian Mathematical Sciences Institute, University of Melbourne*

Time: Tuesday 11-11:40

From the mid 1980â€™s, the author and others adapted integrable nonlinear convection-diffusion equations to obtain realistic one-dimensional solutions for transient unsaturated flow in soil. The solution with constant-flux boundary conditions has been of great interest but the solution with Dirichlet boundary conditions has defied our best efforts. This problem can be transformed to a linear diffusion equation with modified Stefan boundary conditions. If we choose independent coordinates to be canonical coordinates of an approximate scaling symmetry, then separation of variables is admissible at all levels of correction for the non-invariant problem. The full solution is a power series in t1/2 for which remarkably, each term satisfies the governing equation. Using computer algebra, we have now exactly calculated a large number of coefficients for the infiltration series, which is the depth of water having entered the soil, expressed as a power series in t1/2. The results are not those expected from solutions of simpler models but they agree better with experiment.

**Symmetry and mesh adaptivity**

*Christopher Budd*

University of Bath

Department: Mathematics

University of Bath

Department: Mathematics

Time: Wednesday 9-9:40

When solving either PDES or ODEs numerically it is often convenient to use some form of adaptive spatial or temporal mesh. This has the advantage of permitting more accurate computations at reduced computational cost. In this talk I will also show how adaptivity has a natural interpretation in the context of differential equations with symmetry, and indeed that symmetry gives a strong guide as to how an adaptive calculation should proceed. Indeed, I will show that adaptive numerical methods based upon symmetry invariants can give remarkably small computational errors when compared to fixed mesh methods.

**Locally and nonlocally related potential systems. Applications to analysis of PDE systems in fluid and plasma theory.**

*Alexei Cheviakov*

Department of Mathematics and Statistics

University of Saskatchewan

Saskatoon, SK

Department of Mathematics and Statistics

University of Saskatchewan

Saskatoon, SK

Time: Monday 11-11:40

In recent years, many papers have been devoted to analysis of PDE systems using nonlocally related (in particular, potential) PDE systems. Potential systems are obtained by augmenting a given PDE system with potential equations following from an admitted conservation law. The potential variable is normally a nonlocal variable, i.e., cannot be expressed in terms of independent and dependent variables of a given PDE system and their derivatives. Using the nonlocal relation and equivalence of original and potential PDE systems, multiple new analytical results have been obtained for PDE systems arising in various applications. However it turns out that for some PDE systems, conservation laws can yield potential variables that are dependent on local variables of the problem. In particular, this is the case for the axially-symmetric reduction of Euler equations of fluid flow. We demonstrate that the Bragg-Hawthorne equation (Grad-Shafranov equation in MHD theory) is indeed locally related to the original PDE system, and discuss the implications for point symmetry classification of this PDE system. Similar analysis is done for other symmetric reductions of fluid/plasma equations. For a general PDE system, we provide and illustrate a general criterion that can be used to determine whether a particular conservation law yields a local or nonlocal potential variable.

**Practical methods of computation of fluxes of conservation laws. Conservation laws of the G-equation of flame front propagation**

*Alexei Cheviakov*

Department of Mathematics and Statistics

University of Saskatchewan

Saskatoon, SK

Department of Mathematics and Statistics

University of Saskatchewan

Saskatoon, SK

Time: Monday 16:35-16:50

The direct method for finding conservation laws of PDE systems consists in finding conservation law multipliers through determining equations, and a subsequent computation of the corresponding fluxes. The latter often presents a computational challenge. I will discuss four methods of flux computation and illustrate them with examples. A particularly interesting example is the G-equation of flame front propagation.

**Rational Solutions of Soliton Equations and Applications to Vortex Dynamics**

*Peter A. Clarkson*

Institute of Mathematics, Statistics & Actuarial Science,

University of Kent, Canterbury, CT2 7NF, UK

P.A.Clarkson@kent.ac.uk

Institute of Mathematics, Statistics & Actuarial Science,

University of Kent, Canterbury, CT2 7NF, UK

P.A.Clarkson@kent.ac.uk

Time: Friday 11-11:40

In this talk I shall discuss special polynomials associated with rational solutions for the Painleve equations and of the soliton equations which are solvable by the inverse scattering method, including the Korteweg-de Vries, Boussinesq and nonlinear Schrodinger equations.

The Painleve equations are six nonlinear ordinary differential equations that have been the subject of much interest in the past thirty years, which have arisen in a variety of physical applications. Further they may be thought of as nonlinear special functions. Rational solutions of the Painleve equations are expressible in terms of the logarithmic derivative of certain special polynomials. For the second Painleve equation (PII) these polynomials are known as the Yablonskii{Vorob'ev polynomials, first derived in the 1960's by Yablonskii and Vorob'ev. The locations of the roots of these polynomials are shown to have a highly regular triangular structure in the complex plane. The analogous special polynomials associated with rational solutions of the fourth Painleve equation (PIV), which are known as the generalized Hermite polynomials and generalized Okamoto polynomials, are described and it is shown that their roots also have a highly regular structure. The Yablonskii{Vorob'ev polynomials arise in string theory and the generalized Hermite polynomials in the theories of random matrices and orthogonal polynomials.

It is well known that soliton equations have symmetry reductions which reduce them to the Painleve equations, e.g. scaling reductions of the Korteweg-de Vries equation is expressible in terms of PII and scaling reductions of the Boussinesq and nonlinear Schrodinger equations are expressible in terms of PIV. Hence rational solutions of these soliton equations can be expressed in terms of the Yablonskii and Vorob'ev, generalized Hermite and generalized Okamoto polynomials. Further general rational solutions of equations for the Korteweg-de Vries, Boussinesq equations and nonlinear Schrodinger equations, which involve arbitrary parameters, will also be described.

Finally I shall discuss applications of these special polynomials associated with rational solutions for the Painleve and soliton equations to point vortex dynamics.

**The non-autonomous dynamical Lie's systems and exact solutions with superposition principle for evolutionary PDEs**

*Vladimir Dorodnitsyn*

Keldysh Institute of Applied Mathematics of RAS, & Moscow Regional State University, Moscow, Russia; dorod@spp.Keldysh.ru

Keldysh Institute of Applied Mathematics of RAS, & Moscow Regional State University, Moscow, Russia; dorod@spp.Keldysh.ru

Time: Friday 9-9:40

The present talk is devoted to the new application of S. Lie's non-autonomous dynamical systems with the generalized separation of variables in right hand-sides, which possess fundamental sets of particular solutions and nonlinear superposition principles. We consider non-autonomous dynamical equations as some sort of external action on a given evolution PDE. The goal of our approach is to find a subset of solutions of evolution equation which possesses the superposition principle. Here we consider the application of the simplest one-dimensional case of the Lie theorem. The trajectories (solutions) of non-autonomous equation will be considered as some kind of symmetry transformations, which act on evolution equation, transforming a subset of solutions into itself. This leads to integration of ordinary differential equations in a process of finding exact solutions of PDEs. We supply the theory with several examples.

**The Shape of the Optimal Javelin---A dynamical-systems approach using a similarity solution**

*Yossi Farjoun*

MIT

MIT

Time: Tuesday 16:15-16:30

Optimal-shape problems can be often converted into eigenvalue maximization problems. In many cases the resulting ODE is singular at the ends of the domain, and this leads to difficulties in the numerical solution. I will present one such physical problem: finding the taper of the javelin whose lowest mode of vibration has the largest frequency.

The resulting equations are difficult to solve directly and a naive approach fails.

Using a ``similarity solution'' of the ODE, the problem is reduced to a non-linear dynamical-system with a critical point. This new dynamical system is no longer singular and by starting near the critical point and solving the system ``backwards'' the solution is found. The resulting shape has a frequency of vibration 5 times larger than that of the uniform-diameter rod.

The method of solution is applicable to other similar problems. For example, the shape of the tallest column (the problem that inspired this study) can also be found, and with this method the results of J. B. Keller and F. I. Niordson [1] are easily reproduced.

[1] J. B. Keller and F. I. Niordson ; J. Math. Mech. 1966 (16)

**Similarity Solutions for Higher, Odd-order, KdV-type**

*Ray Fernandes*

University of Bath

Department of Mathematics

University of Bath

Department of Mathematics

Time: Wednesday 11:45-12:25

Whilst various even-order PDEs have been well studied and understood, odd-order PDEs pose more difficulties, in particular due to their highly oscillatory nature. We look for similarity solutions, to find the behaviour of the general, odd-order linear PDE $u_t = (-1)^{k+1}D^{2k+1}_x u$, for $k=1,2,\hdots$. Here the case $k=1$ can be shown to be the classic Airy equation. Results found for the linear case can then be used in determining the behavior of related nonlinear equations, such as $u_t =(-1)^{k+1}D^{2k+1}_x(|u|^nu)$. We treat the problems using a variety of techniques including asymptotic analysis, spectral theory and numerical construction.

**Continuous symmetry analysis of dissipative constitutive laws with application to the time-temperature superposition**

*Jean-FranÃ§ois Ganghoffer*

Coll. with Vincent Magnenet

LEMTA â€“ ENSEM, Nancy UniversitÃ©

2, Avenue de la ForÃªt de Haye. BP 160. 54504 Vandoeuvre LÃ¨s Nancy Cedex. France

Coll. with Vincent Magnenet

LEMTA â€“ ENSEM, Nancy UniversitÃ©

2, Avenue de la ForÃªt de Haye. BP 160. 54504 Vandoeuvre LÃ¨s Nancy Cedex. France

Time: Monday 11:45-12:25

The novel aspect advocated in this contribution is the involvement of Lie groups as a predictive method to obtain invariance properties of materials, especially considering inelastic materials such as polymers. The well-known WLF (an acronym for the three authors Williams-Landel-Ferry) approach to condense the materialâ€™s response into so-called master curves is of phenomenological nature; hence, its major drawback lies in fact that the underlying model has to be identified (its coefficients) for each new material, thus generating as many experiments.

An analysis of the continuous symmetries of the constitutive laws of dissipative materials expressed within a thermodynamical framework of relaxation (inspired by the thermodynamics of De Donder, 1936) is performed. This framework relies on the generalization of Gibbâ€™s relationship outside equilibrium and the use of the fluctuation theory developed by Prigogine (1955) to model the internal material dissipation due to its microstructural changes.

An interpretation of the formulated non-equilibrium approach of irreversible processes is given in terms of an extremum principle, associated to a Lagrangian functional. One possible construct for the Lagrangian kernel is the material derivative of the internal energy density, augmented by a complementary term accounting for the evolution laws of the internal variables. Interpreting the material behavior during the non-equilibrium evolution as the Euler-Lagrange equations of the resulting action integral, a differential condition expressing both the local and variational symmetries encapsulated into the Lagrangian formulation is derived. It is further shown that both the variational and the local previous conditions are equivalent along the optimal paths corresponding to the satisfaction of the constitutive laws.

The predictive nature of the symmetry analysis is further highlighted, in terms of both practical and methodological aspects, as a systematic tool for a rational exploitation of the materialâ€™s constitutive response. The performance of the method is exemplified by the construction of a time-temperature equivalence principle associated to the isothermal inelastic behavior of a dry viscous polymer (the polyamid PA66). The calculated shift factor allows condensing the materialâ€™s response into a so-called master curve in the time temperature plane. The predicted shift factor is in good agreement with the empirical shift factor given by the WLF relation. The master curves reflect the invariance properties obtained from the variational symmetries, as articulated in Noetherâ€™s theorem.

From a general point of view, the principal breakthrough of the Lie group analysis in the field of material science lies in its predictive nature, allowing to assess the impact of a variation of the parameters of the material constitutive law, relying on a limited number of experiments.

**SOME MATHEMATICAL PROBLEMS IN BLOOD COAGULATION**

*Miguel A. Herrero*

Departamento de Matematica Aplicada

Facultad de Matematicas

Universidad Complutense

28040 Madrid

Spain

E-mail address . Miguel_Herrero@mat.ucm.es

Departamento de Matematica Aplicada

Facultad de Matematicas

Universidad Complutense

28040 Madrid

Spain

E-mail address . Miguel_Herrero@mat.ucm.es

Time: Wednesday 14:15-14:55

Blood coagulation is a robust security mechanism which prevents bleeding from minor injuries to occur. Any disruption in that system may have serious consequences. Two such extreme situations are illustrated by well-known disorders: haemophilia (which corresponds to low coagulation activity) and disseminated intravascular coagulation, DIC, which is characterized by an excessive activation of coagulation pathways.

In this lecture a mathematical model for blood coagulation will be presented. This model takes into account the interaction of two different modules: an activator-inhibitor one, which triggers the transformation of fibrinogen into the active form fibrin, and a polymerization one that accounts for the formation of fibrin aggregates. These last provide the basic scaffold for the formation of blood clots, whose early stages are characterized by the onset of microthrombi clouds. We shall describe some properties of the system under consideration, and in particular we shall discuss on the relation between the location of induced microthrombi clouds and the size of the activation source when this last is located at a tissue adjacent to the vessel wall, as it often happens in a number of pathological situations.

The work to be reported upon has been done in collaboration with G. Guria and K. Zlobina, from the National Center for Haematology at Moscow (Russia).

**Computer Algebra Implementation of Symmetry Classification with Invariance**

*Tracy Huang,*

Faculty of Information Sciences & Engineering, University of Canberra

Faculty of Information Sciences & Engineering, University of Canberra

Time: Wednesday 16:15-16:30

The symmetry classification problem for PDEs can be approached algorithmically by applying differential reduction and completion to the symmetry determining equations. Splitting conditions should be invariant under the action of the equivalence group for the PDEs. We provide code in the computer algebra language Maple for finding determining equations of the equivalence group. By modifying the pivot selection rules of Reid and Wittkopf's 'rif' algorithm, we give a Maple implementation for symmetry classification with preferential selection of invariant pivots. Examples are presented to illustrate the implementation.

Generating Sets of Conservation Laws and Equivalence of Potential Systems

Generating Sets of Conservation Laws and Equivalence of Potential Systems

*Nataliya Ivanova*

Institute of Mathematics

Kyiv, Ukraine

Institute of Mathematics

Kyiv, Ukraine

Time: Thursday 11-11:40

We introduce the notions of generating sets of conservation laws of systems of differential equations with respect to symmetry groups and equivalence groups. The notion of equivalence of conservation laws with respect to a group of transformations is generalized in several directions: classification of pairs ``system + space of conservation laws''; classification of conservation laws for a given system with respect to its symmetry group; classification of pairs ``system + a conservation law''. This allows us to generalize essentially the procedure of finding potential symmetries for the systems with multidimensional spaces of conservation laws. Namely, previously, for construction of simplest potential systems in cases when the dimension of the space of local conservation laws is greater than one, only basis conservation laws were used. However, the basis conservation laws may be equivalent with respect to groups of symmetry transformations, or vice versa, the number of $G^{\sim}$-independent linear combinations of conservation laws may be grater then dimension of the space of conservation laws. The first possibility leads to an unnecessary, often cumbersome, investigation of equivalent systems, the second one makes possible missing a great number of inequivalent potential systems. Below we show how to choose conservation laws in order to obtain all possible inequivalent potential systems associated to the given system.

**Self-similar behaviour in 'fast' nonlinear diffusion**

*John King*

University of Nottingham

Department of Mathematical Sciences

University of Nottingham

Department of Mathematical Sciences

Time: Tuesday 9:45-10:25

The asymptotic behaviour of various initial-boundary-value problems for the equation of fast nonlinear diffusion will be described, with emphasis placed on (i) the various mechanisms that can be responsible for selecting the relevant similarity exponent and (ii) the role played by the critical case associated with Yamabe flow.

**Algorithmic Testing of Invariance in Classifying Symmetries of PDE**

*Ian Lisle,*

Faculty of Information Sciences & Engineering, University of Canberra

Faculty of Information Sciences & Engineering, University of Canberra

Time: Wednesday 15-15:40

Differential reduction algorithms can be applied to solve the symmetry classification problem for PDEs. The most important property of the resulting classes is that they respect the action of the equivalence group for the PDEs, so that equations known to be connected by a change of variable are grouped in the same class. We demonstrate how to test the invariance of candidate case splittings algorithmically. The method uses only the infinitesimal determining equations for symmetries and equivalences: an explicit parameterisation of the equivalence group is not required.

**Similarity Solutions and Traveling Waves in Dewetting Liquid Films with Large Slip**

*Andreas MuenchUniversity of NottinghamDepartment of Mathematical SciencesTime: Thursday 9-9:40*

In this talk we consider situations where liquid films dewet from a solid surface with a large effective slip at the liquid solid boundary. In the first situation, a film ruptures under the influence of intermolecular forces. Near rupture time, the solution to the thin film model passes through several self-similar regimes including one of second kind. In this regime, the equation can be reduced to a mean-field type equation by the introduction of Lagrangian coordinates.

If time permits, we will include other stages of dewetting processes for example one where inertia is dominant. Here, the rapid dewetting can give rise to double wave solutions which correspond to shock solutions, including non-classical shock solutions that violate the Lax condition.

**The importance of inheriting the Lie symmetry algebra.**

*MC Nucci*

Universita di Perugia (Italy)

Universita di Perugia (Italy)

Time: Tuesday 14:15-14:55

We underline the importance of Lie classical and nonclassical symmetries in finding solutions of partial differential equations (PDE). We present the method of iterating the nonclassical symmetries. It produces new nonlinear equations, which inherit the Lie point symmetry algebra of the given equation. It has already been shown that conditional symmetries can be indeed retrieved by solutions of the heir-equations. In particular we exemplify that: (1) one can recover solutions of a PDE -- which cannot be found by either classical or nonclassical symmetries -- by means of a Lie symmetry solution of the heir-equations; (2) special solutions of the right-order heir-equation correspond to classical and nonclassical symmetries of the original equations.

References:

M.C. Nucci, "Iterating the nonclassical symmetries method"

Physica D 78 (1994) pp. 124--134.

M.C. Nucci, "Iterations of the nonclassical symmetries and conditional

Lie-BÃ¤cklund symmetries", J. Phys. A: Math. Gen. 29 (1996) pp. 8117--8122.

M.C. Nucci, "Nonclassical symmetries as special solutions of heir-equations"

J. Math. Anal. Appl. 279 pp. 168-179 (2003)

S. Martini, N. Ciccoli and M.C. Nucci, "Group analysis and heir-equations of a mathematical model for thin liquid films" J. Nonlinear Math. Phys. (2008) to appear

**Lagrangians for biological systems.**

*MC Nucci*

Universita di Perugia (Italy)

Universita di Perugia (Italy)

Time: Wednesday 16:35-16:50

We show that it is possible to find Lagrangians for system of ordinary differential equation, even for system of first order by using the method of the Jacobi last multiplier and the connection with Lie symmetries which was found by Lie himself. Several examples drawn from biology are presented.

References:

M.C. Nucci and A.M. Arthurs ``On the inverse problem of calculus of variations" (2008)

M.C. Nucci and K.M. Tamizhmani, ``An old method by Jacobi to derive Lagrangians for a nonlinear dynamical system with variable coefficients" (2008)

M.C. Nucci and K.M. Tamizhmani, ``Lagrangians for biological models: the use of Jacobi last multiplier and Lie symmetries" (2008)

**Generalized symmetries of the G-equation for premixed combustion and high order accurate numerical ray-tracing schemes**

*Martin Oberlack*

Technische UniversitÃ¤t Darmstadt

Technische UniversitÃ¤t Darmstadt

Time: Monday 16:15-16:30

It is shown that the admissible symmetries of the G-equation for flame front propagation of premixed combustion are considerable extended if generalized symmetries are considered. Classical symmetries are exhaustively discussed in (Oberlack, Wenzel, Peters 2001}. If the flow velocity is zero, an infinite series of generalized (Lie-BÃ¤cklund) symmetries has been derived (Oberlack 2004). Presently it is shown that the G-equation also admits an infinite sequence of symmetries for a non-zero velocity field. The first elements up to second order derivatives have been computed. Geometrical and kinematic interpretations of the symmetries are given. Finally it is shown that higher order symmetries may be used to derive numerical ray tracing schemes up to arbitrary precision.

**On a new Lie-symmetry of the multi-point equation for turbulence and its implications for fractal generated turbulence**

*Martin Oberlack*

Technische UniversitÃ¤t Darmstadt

Technische UniversitÃ¤t Darmstadt

Time: Wednesday 9:45-10:25

Investigating the multi-point correlation equations for the velocity and pressure fluctuations in the limit of homogeneous turbulence a new scaling symmetry has been discovered. Interesting enough this property is not shared with the Euler or Navier-Stokes equations from which the multi-point correlation equations have originally emerged. This was first observed for parallel wall-bounded shear flows (Khujadze, Oberlack 2004) though there this property only holds true for the two-point equation. Hence, in a strict sense there it is broken for higher order correlation equations. Presently using this extended set of symmetry groups a much wider class of invariant solutions or turbulent scaling laws is derived for homogeneous turbulence. In particular, we show that the experimentally observed specific scaling properties of fractal-generated turbulence (Vassilicos) fall into this new class of solutions. This is in particular a constant integral and Taylor length scale downstream of the fractal grid and the exponential decay of the turbulent kinetic energy along the same axis. These particular properties can only be conceived from multi-point equations using the new scaling symmetry since the two classical scaling groups of space and time are broken for this specific case. Hence, extended statistical scaling properties going beyond the Euler and Navier-Stokes have been clearly observed in experiments for the first time.

**Similarity in Ray Theory**

*John Ockendon*

University of Oxford

University of Oxford

Time: Monday 14:15-14:55

This talk will give a brief review of some of the similarity reductions that can give interesting physical insights into various diffraction phenomena.

**Meshfree solver for scalar conservation laws in one space dimension**

*Benjamin Seibold*

MIT

MIT

Time: Wednesday 11-11:40

A meshfree numerical solver for scalar conservation laws in one space dimension is presented. Particles representing the solution are moved according to their characteristic velocities in is usual for particle methods. By using a similarity solution to interpolate between the particles, a ``natural'' description of the area under the solution is found. This allows merging particles while conserving the area. Since no global remeshing is done, shocks stay sharp and the solution away from shocks is unperturbed. The conservation of area guarantees that shocks propagate at the correct speed, and rarefaction waves are created where appropriate. The method is TVD, entropy decreasing, exactly conservative, and has no numerical dissipation. Transonic points do not pose any special difficulty, however inflection points of the flux function pose a slight challenge that can be overcome by a special treatment. Away from shocks the method is second order accurate, while shocks are resolved with first order accuracy. A post processing step can recover the second order accuracy for shocks as well. The method is compared to CLAWPACK in test cases and is found to yield an increase in accuracy for comparable resolutions. The method can also be used when there is a source term. In this case the solution can either be found be operator splitting, or by solving an ODE for the value and location of each particle.

**Group analysis of variable coefficient nonlinear wave equations**

Lina Song

Department of Applied Mathematics

Dalian University of Technology, Dalian 116024, P. R. China

Lina Song

Department of Applied Mathematics

Dalian University of Technology, Dalian 116024, P. R. China

Time: Tuesday 15-15:40

A group classification of a class of (1+1)-dimensional variable coefficient nonlinear wave equations f(x)utt = (k(x)F(u)ux)x + g(x)H(u)ux + d(x)K(u)ut is given. A number of new interesting nonlinear invariant models which have nontrivial invariance algebras are obtained. Furthermore, exact solutions of special forms of the equations are also constructed via classical Lie method and generalized conditional transformations. The multipliers yielding local conservation laws with characteristics of order 0 of the class under consideration are classified with respect to the group of equivalence transformations. The results of the work are summarized in Table I-V.

Â¤The work is partially supported by the National Key Basic Research Project of China under the Grant

NO.2004CB318000.

E-mail address: songlina1981@yahoo.com.cn

**Differential Characteristic Set Algorithm for the Complete Symmetry Classification of (Partial) Differential equations**

*Temuerchaolu*

Maritime University, Shanghai

Maritime University, Shanghai

Time: Thursday 11:45-12:25

In this talk, a differential polynomial characteristic set algorithm for the complete symmetry classification of (partial) differential equations with some parameters is given, which makes the solution of the complete symmetry classification problem for (partial) differential equations become direct and systematic. As an illustrative example, the complete classical and potential symmetry classifications of the wave equation with an arbitrary function parameter are presented. This is a new application of the differential form characteristic set algorithm (differential form Wuâ€™s method) in differential fields.

**Singularity formation in the harmonic map heat flow**

*Jan Bouwe van den Berg*

VU University, Amsterdam

VU University, Amsterdam

Time: Monday 9:45-10:25

The harmonic map heat flow is a model for nematic liquid crystals and also has origins in geometry. We will introduce the model and discuss some of its mathematical properties. In particular, we will focus on the possibility that singularities may develop. The rate at which singularities develop is investigated in settings with certain symmetries. We use the method of matched asymptotic expansions and identify different scenarios for singularity formation. More specifically, we distinguish between singularities that develop in finite time and those that need infinite time to form. Finally, we discuss which results can be proven rigorously, as well as some open problems, and we address stability issues (ongoing work with JF Williams).

**Coarsening dynamics of slipping droplets**

*Barbara WagnerWeierstrass Institute for Applied Analysis and Stochastics, Berlin*

* *

Time: Thursday 9:45-10:25

In this talk the late phase dewetting process of nanoscopic thin polymer films on hydrophobized substrates using some recently derived lubrication models that take account of large slippage at the polymer-substrate interface are presented. The late phase of this process is characterized by the slow-time coarsening dynamics of arrays of droplets that remain after rupture and the initial dewetting phases. For this situation a reduced system of ordinary differential equations is derived from the lubrication model for large slippage using asymptotic analysis. This extends known results for the no-slip case. On the basis of the reduced model, the role of the slippage as a control parameter for droplet migration is analyzed and three new qualitative effects for the coarsening process are identified.**Adaptive numerical methods for PDE with singularities***JF Williams, Simon Fraser University*

Time: Friday 9:45-10:25

This talk will describe scale-invariant adaptive strategies in both time and space for the reliable resolution of finite-time singularities. Both strategies are based on preserving the underlying symmetries of the physical PDE. Examples in one, two and three dimensions will be present.**On similarity solutions for fluid film rupture***Thomas WitelskiDuke University*

Time: Tuesday 9-9:40

Finite-time topological rupture occurs in many models in fluid and solid mechanics. We review and discuss some properties of the self-similar solutions for such problems. Unresolved issues regarding analytical forms of the solutions (stability and symmetry vs. asymmetry) and numerical calculation methods (shooting vs. global relaxation) will be highlighted. Progress on describing the set of solutions for van der Waals driven thin film rupture via beyond all orders asymptotic will be outlined.

**Demo of the computer algebra programs Lie PDE, ConLaw, ApplySym and Crack for computing and applying symmetries and conservation laws**

*Thomas Wolf*

Brock University

Brock University

Time: Monday 15-15:40

In the demo special features of the package Crack are demonstrated, like the ability to do interactively point transformations, to work with non-polynomial equations that have unknowns in exponents and recent tools to work with very large expressions including parallelization. Systems that have been investigated recently are the equations describing one-dimensional gas dynamics, isentropic and non-isentropic.

**Singularity Structure Analysis and similarity reductions for the (2+1)-Dimensional Generalized Burgers Equation**

*Zhenya Yan*

Chinese Academy of Sciences, Beijing

Chinese Academy of Sciences, Beijing

Time: Tuesday 16:55-17:35

In this talk, the singularity structure analysis is performed for the (2+1)-dimensional generalized Burgers equation where it is shown that the equation passes the Painleve test and a Backlund transformation is also obtained. Moreover, a direct method is used to obtain similarity reductions and conditional similarity reductions of the (2+1) dimensional generalized Burgers equation such that eleven families of similarity reductions and six families of conditional similarity reductions of the equation are found.

## Details

The aim of this conference is to bring together researchers who apply similarity analysis to engineering problems, researchers who are developing significant extensions of similarity methods so that they can be applied more widely, numerical analysts who develop and use similarity methods in their numerical schemes and software researchers who develop symbolic manipulation software to implement similarity analysis. It is intended that one afternoon will be devoted to discussion of open problems related to the objectives of the conference. The conference will begin on the morning of August 11th and end on the afternoon of August 15th. The abstracts and titles of the talks of the confirmed participants should be sent to either of the local organizers, George Bluman or Michael Ward, by June 15th.

There is some limited funding for travel and accommodation for those
graduate students and postdoctoral fellows who work in this area and
who would like to attend the conference and present a poster. Please
contact either of the local organizers, George Bluman or Michael Ward, for further details.

## Additional Information

**Accommodation**

A block of rooms will be reserved for confirmed participants. For confirmed participants, it is important to register for the conference before June 1st in order to guarantee your room.

**Participants**

Anco, Stephen (Brock University)

Bernoff, Andrew (Harvey Mudd College)

Bertozzi, Andrea (UCLA)

Bluman, George (UBC)

Bogoyavlenskij, Oleg (Queen's University)

Broadbridge, Phil (Australian Mathematical Sciences Institute)

Budd, Christopher (University of Bath)

Cheviakov, Alexei F (University of Saskatchewan)

Clarkson, Peter (University of Kent)

Dorodnitsyn, Vladimir Keldysh (Institute of Applied Mathematics &

Moscow Regional State University)

Farjoun, Yossi (MIT)

Fernandes, Ray (University of Bath)

Ganghoffer, Jean-FranÃ§ois (LEMTA)

Herrero, Miguel Angel (Universidad Complutense)

Huang, Shih-lung Tracy (University of Canberra)

Ivanova, Nataliya (Institute of Mathematics of NAS of Ukraine)

King, John R. (University of Nottingham)

Lisle, Ian (University of Canberra)

Mansfield, Elizabeth (University of Kent)

Muench, Andreas (University of Nottingham)

Nucci, Maria Clara (Universita di Perugia)

Oberlack, Martin Technische (UniversitÃ¤t Darmstadt)

Ockendon, John (Oxford University)

Pugh, Mary (University of Toronto)

Seibold, Benjamin (MIT)

Song, Lina Dalian (University of Technology)

Stockie, John (Simon Fraser University)

Temuer, Chaolu (Maritime University)

van den Berg, Jan Bouwe (VU University Amsterdam)

Wagner, Barbara Weierstrauss (Institute for Applied Analysis and Stochastics)

Ward, Michael (UBC)

Williams, JF (Simon Fraser University)

Witelski, Thomas (Duke University)

Wolf, Thomas (Brock University)

Yan, Zhenya (Chinese Academy of Sciences)

*(Group photo of conference participants)*

See Below

**Scientific, Seminar**

**August 11â€“15, 2008**

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