One-parameter Family Of Solutions Research Articles

Solutions of the porous medium equation with degenerate interfaces

We prove the existence of a one-parameter family of solutions of the porous medium equation in which the interface is a half line whose end point advances at a constant speed. Then we prove the stability of the solutions under a suitable class of perturbations. We discuss the relevance of these solutions to gravity-driven flows of thin films, and show that some solutions develop a very thin triangular plateau in the direction of propagation and that the angle of the plateau and its thickness are decreasing functions of the speed.

The flow in a divergent channel driven by stretching the inclined walls

We consider the steady two-dimensional flow of an incompressible viscous fluid between the walls of a divergent channel; the walls are assumed to be stretching, and further, they meet at an angle 2α, where 0 < α ⩽ π. A similarity solution is sought and a Reynolds number, R, based on the motion of the walls is introduced. The apparently simple case when R = 0 is discussed but its relevance is only found later. For general R the form of the solution results in a cubic equation with the discriminant Δ. Three cases arise: (i) Δ = 0, (ii) Δ < 0 and (iii) Δ > 0. It is found that in the case (i) there is a one-parameter family of solutions. In the cases (ii) and (iii) solutions are found in terms of Jacobian elliptic functions with parameter m. In the case (ii) for each value of m a one-parameter family of solutions is again found. In case (iii) there are multiple solutions in the interval 0 < α ⩽ π. Furthermore, from a small-m analysis it is found that the solution in the case when R = 0 is dependent on the nonlinear inertia terms. All the solutions found are symmetric.

Can quantum effects due to a massless conformally coupled field avoid gravitational singularities?

Using quantum corrections from massless fields conformally coupled to gravity, we study the possibility of avoiding singularities that appear in the flat Friedmann-Robertson-Walker model. We assume that the universe contains a barotropic perfect fluid with the state equation p = ωρ, where p is the pressure and ρ is the energy density. We study the dynamics of the model for all values of the parameter ω and also for all values of the conformal anomaly coefficients α and β. We show that singularities can be avoided only in the case where α > 0 and β −1 (only a one-parameter family of solutions, but no a general solution, has this behavior at late times), the initial conditions of the nonsingular solutions at early times must be chosen very exactly. These nonsingular solutions consist of a general solution (a two-parameter family) exiting the contracting de Sitter phase and a one-parameter family exiting the contracting Friedmann phase. On the other hand, for ω < −1 (a phantom field), the problem of avoiding singularities is more involved because if we consider an expanding Friedmann phase at early times, then in addition to fine-tuning the initial conditions, we must also fine-tune the parameters α and β to obtain a behavior without future singularities: only a oneparameter family of solutions follows a contracting Friedmann phase at late times, and only a particular solution behaves like a contracting de Sitter universe. The other solutions have future singularities.

Braidings of Tensor Spaces

Let V be a braided vector space, i.e., a vector space together with a solution \({\hat{R}\in {{End}}(V\otimes V)}\) of the Yang–Baxter equation. Denote \({T(V):=\bigoplus_k V^{\otimes k}}\) . We associate to \({\hat{R}}\) a one-parameter family of solutions \({T(\hat{R})\in {\rm End}(T(V)\otimes T(V))}\) of the Yang–Baxter equation on the tensor space T (V). Main ingredients of the solution are braid analogues of the binomial coefficients and of the Pochhammer symbols. The association \({\hat{R}\rightsquigarrow T(\hat{R})}\) is functorial with respect to V.

Quantum critical superfluid flows and anisotropic domain walls

We construct charged anisotropic AdS domain walls as solutions of a consistent truncation of type IIB string theory. These are a one-parameter family of solutions that flow to an AdS fixed point in the IR, exhibiting emergent conformal invariance and quantum criticality. They represent the zero-temperature limit of the holographic superfluids at finite superfluid velocity constructed in arXiv:1010.5777. We show that these domain walls exist only for velocities less than a critical value, agreeing in detail with a conjecture made there. We also comment about the IR limits of flows with velocities higher than this critical value, and point out an intriguing similarity between the phase diagrams of holographic superfluid flows and those of ordinary superconductors with imbalanced chemical potential.

On Matrix-Free Computation of 2D Unstable Manifolds

Recently, a flexible and stable algorithm was introduced for the computation of two-dimensional (2D) unstable manifolds of periodic solutions to systems of ordinary differential equations. The main idea of this approach is to represent orbits in this manifold as the solutions of an appropriate boundary value problem (BVP). The BVP is underdetermined, and a one-parameter family of solutions can be found by means of arclength continuation. This family of orbits covers a piece of the manifold. The quality of this covering depends on the way the BVP is discretized, as do the tractability and accuracy of the computation. In this paper, we describe an implementation of the orbit continuation algorithm which relies on multiple shooting and Newton–Krylov continuation. We show that the number of time integrations necessary for each continuation step scales with the number of shooting intervals but not with the number of degrees of freedom of the dynamical system. The number of shooting intervals is chosen based on linear stability analysis to keep the conditioning of the BVP in check. We demonstrate our algorithm with two test systems: a low-order model of shear flow and a well-resolved simulation of turbulent plane Couette flow.

Vorticesin(2+1)d conformal fluids

We study isolated, stationary, axially symmetric vortex solutions in (2+1)-dimensional viscous conformal fluids. The equations describing them can be brought to the form of three coupled first order ODEs for the radial and rotational velocities and the temperature. They have a rich space of solutions characterized by the radial energy and angular momentum fluxes. We do a detailed study of the phases in the one-parameter family of solutions with no energy flux. This parameter is the product of the asymptotic vorticity and temperature. When it is large, the radial fluid velocity reaches the speed of light at a finite inner radius. When it is below a critical value, the velocity is everywhere bounded, but at the origin there is a discontinuity. We comment on turbulence, potential gravity duals, non-viscous limits and non-relativistic limits.

One-parameter family of solutions for one class of hammerstein nonlinear equations on a half-line

One-parameter family of solutions for one class of hammerstein nonlinear equations on a half-line

DOMAIN WALL SOLITONS AND HOPF ALGEBRAIC TRANSLATIONAL SYMMETRIES IN NONCOMMUTATIVE FIELD THEORIES

Domain wall solitons are the simplest topological objects in field theories. The conventional translational symmetry in a field theory is the generator of a one-parameter family of domain wall solutions, and induces a massless moduli field which propagates along a domain wall. We study similar issues in braided noncommutative field theories possessing Hopf algebraic translational symmetries. As a concrete example, we discuss a domain wall soliton in the scalar braided noncommutative field theory in Lie-algebraic noncommutative spacetime, which has a Hopf algebraic translational symmetry. We construct explicitly a one-parameter family of solutions in perturbation of the noncommutativity parameter. We then find the massless moduli field which propagates on the domain wall soliton. This work is based on arXiv:0711.3059.

Domain wall solitons and Hopf algebraic translational symmetries in noncommutative field theories

Domain wall solitons are the simplest topological objects in field theories. The conventional translational symmetry in a field theory is the generator of a one-parameter family of domain wall solutions, and induces a massless moduli field which propagates along a domain wall. We study similar issues in braided noncommutative field theories possessing Hopf algebraic translational symmetries. As a concrete example, we discuss a domain wall soliton in the scalar ${\ensuremath{\phi}}^{4}$ braided noncommutative field theory in Lie-algebraic noncommutative space-time, $[{x}^{i},{x}^{j}]=2i\ensuremath{\kappa}{ϵ}^{ijk}{x}_{k}$ $(i,j,k=1,2,3)$, which has a Hopf algebraic translational symmetry. We first discuss the existence of a domain wall soliton in view of Derrick's theorem, and construct explicitly a one-parameter family of solutions in perturbation of the noncommutativity parameter $\ensuremath{\kappa}$. We then find the massless moduli field which propagates on the domain wall soliton. We further extend our analysis to the general Hopf algebraic translational symmetry.

Lagrangian formulation of a generalized Lane-Emden equation and double reduction

We classify the Noether point symmetries of the generalized Lane-Emden equation y″+ ny′/x+ f(y) = 0 with respect to the standard Lagrangian L = xny′2/2 — xn ∫f(y)dy for various functions f(y). We obtain first integrals of the various cases which admit Noether point symmetry and find reduction to quadratures for these cases. Three new cases are found for the function f(y). One of them is f(y) = αyr , where r ≠ 0,1. The case r = 5 was considered previously and only a one-parameter family of solutions was presented. Here we provide a complete integration not only for r = 5 but for other r values. We also give the Lie point symmetries for each case. In two of the new cases, the single Noether symmetry is also the only Lie point symmetry.

Gravity-driven thin liquid films with insoluble surfactant: smooth traveling waves

The flow of a thin layer of fluid down an inclined plane is modified by the presence of insoluble surfactant. For any finite surfactant mass, traveling waves are constructed for a system of lubrication equations describing the evolution of the free-surface fluid height and the surfactant concentration. The one-parameter family of solutions is investigated using perturbation theory with three small parameters: the coefficient of surface tension, the surfactant diffusivity, and the coefficient of the gravity-driven diffusive spreading of the fluid. When all three parameters are zero, the nonlinear PDE system is hyperbolic/degenerate-parabolic, and admits traveling wave solutions in which the free-surface height is piecewise constant, and the surfactant concentration is piecewise linear and continuous. The jumps and corners in the traveling waves are regularized when the small parameters are nonzero; their structure is revealed through a combination of analysis and numerical simulation.

One-parameter localized traveling waves in nonlinear Schrödinger lattices

We address the existence of traveling single-humped localized solutions in the spatially discrete nonlinear Schrödinger (NLS) equation. A mathematical technique is developed for analysis of persistence of these solutions from a certain limit in which the dispersion relation of linear waves contains a triple zero. The technique is based on using the Implicit Function Theorem for solution of an appropriate differential advance–delay equation in exponentially weighted spaces. The resulting Melnikov calculation relies on a number of assumptions on the spectrum of the linearization around the pulse, which are checked numerically. We apply the technique to the so-called Salerno model and the translationally invariant discrete NLS equation with a cubic nonlinearity. We show that the traveling solutions terminate in the Salerno model whereas they generally persist in the translationally invariant NLS lattice as a one-parameter family of solutions. These results are found to be in a close correspondence with numerical approximations of traveling solutions with zero radiation tails. Analysis of persistence also predicts the spectral stability of the one-parameter family of traveling solutions under time evolution of the discrete NLS equation.

Asymptotic expansions for axially symmetric cavities

The paper presents the results of an asymptotic theory of axially symmetric cavity flows at small and zero cavitation number. The results have been obtained on the basis of the variational Riabouchinsky principle and the asymptotic theory of slender bodies. This variational-asymptotic approach has been applied to deduce asymptotic expansions for the shape of the cavity and the force exerted on the cavitator at small cavitation number. At the zero cavitation number an integro-differential equation for the shape of the free streamline has been obtained. An exact integral of the equation has been found and a one-parameter family of solutions has been constructed and which has refined earlier asymptotics of Levinson and Gurevich. The equation and asymptotic expansion are independent of the cavitator shape and in this sense are maximally accurate. Any further amendment of the equation of higher order of accuracy would be connected with the shape of the cavitator.

Theoretical skin-friction law in a turbulent boundary layer

We study transitional and turbulent boundary layers using a turbulent velocity profile equation recently derived from the Navier–Stokes-alpha and Leray-alpha models. From this equation we obtain a theoretical prediction of the skin-friction coefficient in a wide range of Reynolds numbers based on momentum thickness, and deduce the maximal value of c f max = 0.0063 for turbulent velocity profiles. A two-parameter family of solutions to the equation matches experimental data in the transitional boundary layers with different free-stream turbulence intensity, while one-parameter family of solutions, obtained using our skin-friction coefficient law, matches experimental data in the turbulent boundary layer for moderately large Reynolds numbers.

The Baryonic Branch of Klebanov-Strassler Solution: a Supersymmetric Family of SU(3) Structure Backgrounds

We exhibit a one-parameter family of regular supersymmetric solutions of type IIB theory that interpolates between Klebanov-Strassler (KS) and Maldacena-Nunez (MN). The solution is obtained by applying the supersymmetry conditions for SU(3)-structure manifolds to an interpolating ansatz proposed by Papadopoulos and Tseytlin. Other than at the KS point, the family does not have a conformally-Ricci-flat metric, neither it has self-dual three-form flux. Nevertheless, the asymptotic IR and UV are that of KS troughout the family, except for the extremal value of the interpolating parameter where the UV solution drastically changes to MN. This one-parameter family of solutions is interpreted as the dual of the baryonic branch of gauge theory, confirming the expecation that the KS solution corresponds to a particular symmetric point in the branch.

Deformations of Frobenius structures on Hurwitz spaces

Deformations of Dubrovin's Hurwitz Frobenius manifolds are constructed. The deformations depend on $g(g+1)/2$ complex parameters where $g$ is the genus of the corresponding Riemann surface. In genus one, the flat metric of the deformed Frobenius manifold coincides with a metric associated with a one-parameter family of solutions to the Painlev\'e-VI equation with coefficients $(1/8,-1/8,1/8,3/8).$ Analogous deformations of the real doubles of the Hurwitz Frobenius manifolds are also found; these deformations depend on $g(g+1)/2$ real parameters.

A free boundary problem for a singular system of differential equations: An application to a model of tumor growth

In this paper we consider a free boundary problem for a nonlinear system of two ordinary differential equations, one of which is singular at some points, including the initial point r = 0 r=0 . Because of the singularity at r = 0 r=0 , the initial value problem has a one-parameter family of solutions. We prove that there exists a unique solution to the free boundary problem. The proof of existence employs two “shooting” parameters. Analysis of the profiles of solutions of the initial value problem and tools such as comparison theorems and weak limits of solutions play an important role in the proof. The system considered here is motivated by a model in tumor growth, but the methods developed should be applicable to more general systems.

Bright compact breathers.

In this communication we will consider the potential of some general classes of nonlinear lattice models to support bright discrete compact breather solutions (compactlets). We analyze the conditions for which such solutions are possible and classify the models as belonging in three general categories: a class with no compact breather solutions, one with one-parameter families of solutions, and a class with "isolated" solutions (i.e., no free parameters). In the latter two cases we construct the solutions and analyze their linear stability. The drastically different stability features of these solutions in comparison with their smoothly decaying counterparts are discussed. Stable breather solutions with compact support are identified in the one-parameter families of solutions, while the corresponding solutions found in the zero-parameter families are always found to be unstable.

Letter: Generation of Solutions of the Einstein Equations by Means of the Kaluza–Klein Formulation

It is shown that starting from a solution of the Einstein–Maxwell equations coupled to a scalar field given by the Kaluza–Klein theory, invariant under a one-parameter group, one can obtain a one-parameter family of solutions of the same equations.