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Jean Dolbeault - One of the best experts on this subject based on the ideXlab platform.

  • branches of non symmetric critical points and symmetry breaking in nonlinear elliptic partial differential equations
    Nonlinearity, 2014
    Co-Authors: Jean Dolbeault, Maria J Esteban
    Abstract:

    In this paper we study the Bifurcation of branches of non-symmetric solutions from the symmetric branch of solutions to the Euler-Lagrange equations satisfied by optimal functions in functional inequalities of Caffarelli-Kohn-Nirenberg type. We establish the asymptotic behavior of the branch for large values of the Bifurcation Parameter. We also perform a formal expansion in a neighborhood of the first Bifurcation point on the branch of symmetric solutions, that characterizes the local behavior of the non-symmetric branch. These results are compatible with earlier numerical and theoretical observations. Further numerical results allow us to distinguish two global scenarii. This sheds a new light on the symmetry breaking phenomenon.

  • Branches of non-symmetric critical points and symmetry breaking in nonlinear elliptic partial differential equations
    Nonlinearity, 2014
    Co-Authors: Jean Dolbeault, Maria J Esteban
    Abstract:

    In this paper we study the Bifurcation of branches of non-symmetric solutions from the symmetric branch of solutions to the Euler-Lagrange equations satisfied by optimal functions in functional inequalities of Caffarelli-Kohn-Nirenberg type. We establish the asymptotic behavior of the branches for large values of the Bifurcation Parameter. We also perform an expansion in a neighborhood of the first Bifurcation point on the branch of symmetric solutions, that characterizes the local behavior of the non-symmetric branch. These results are compatible with earlier numerical and theoretical observations. Further numerical results allow us to distinguish two global scenarios. This sheds a new light on the symmetry breaking phenomenon.

  • Non-Existence and Uniqueness Results for Supercritical Semilinear Elliptic Equations
    Annales Henri Poincaré, 2010
    Co-Authors: Jean Dolbeault, Robert Stańczy
    Abstract:

    Non-existence and uniqueness results are proved for several local and non-local supercritical Bifurcation problems involving a semilinear elliptic equation depending on a Parameter. The domain is star-shaped and such that a Poincaré inequality holds but no other symmetry assumption is required. Uniqueness holds when the Bifurcation Parameter is in a certain range. Our approach can be seen, in some cases, as an extension of non-existence results for non-trivial solutions. It is based on Rellich–Pohožaev type estimates. Semilinear elliptic equations naturally arise in many applications, for instance in astrophysics, hydrodynamics or thermodynamics. We simplify the proof of earlier results by K. Schmitt and R. Schaaf in the so-called local multiplicative case, extend them to the case of a non-local dependence on the Bifurcation Parameter and to the additive case, both in local and non-local settings.

  • Non-existence and uniqueness results for supercritical semilinear elliptic equations
    Annales Henri Poincaré, 2009
    Co-Authors: Jean Dolbeault, Robert Stanczy
    Abstract:

    Non-existence and uniqueness results are proved for several local and non-local supercritical Bifurcation problems involving a semilinear elliptic equation depending on a Parameter. The domain is star-shaped but no other symmetry assumption is required. Uniqueness holds when the Bifurcation Parameter is in a certain range. Our approach can be seen, in some cases, as an extension of non-existence results for non-trivial solutions. It is based on Rellich-Pohozaev type estimates. Semilinear elliptic equations naturally arise in many applications, for instance in astrophysics, hydrodynamics or thermodynamics. We simplify the proof of earlier results by K. Schmitt and R. Schaaf in the so-called local multiplicative case, extend them to the case of a non-local dependence on the Bifurcation Parameter and to the additive case, both in local and non-local settings.

Zhengdong Cheng - One of the best experts on this subject based on the ideXlab platform.

  • target wave to spiral wave pattern transition in a discrete belousov zhabotinsky reaction driven by inactive resin beads
    Physical Review E, 2010
    Co-Authors: Guanqun Wang, Qingsheng Wang, Srinivasa Rao Pullela, Manuel Marquez, Zhengdong Cheng
    Abstract:

    Wave pattern formation and transition in chemical and biochemical reaction systems can reveal the system properties. We investigate the pattern transition from target waves to spiral waves upon the increment of inactive beads in a discrete system model, where ion-exchange resin loaded with Belousov-Zhabotinsky catalyst corresponds to the active beads. The results show that inactive beads slow the propagation speed of target waves and increase the wave frequency. As the population of inactive beads increases, clusters are formed, which then break waves into segments where bounded spiral pairs are generated and separated into individual spirals. From this observation, we conclude that the population of inactive resin beads acts as the Bifurcation Parameter controlling the wave pattern transition from targets to spirals.

Pavel Plotnikov - One of the best experts on this subject based on the ideXlab platform.

  • Small divisor problem in the theory of three-dimensional water gravity waves
    Memoirs of the American Mathematical Society, 2009
    Co-Authors: Gérard Iooss, Pavel Plotnikov
    Abstract:

    We consider doubly-periodic travelling waves at the surface of an infinitely deep perfect fluid, only subjected to gravity g and resulting from the nonlinear interaction of two simply periodic travelling waves making an angle 2θ between them. Denoting by µ = gL/c 2 the dimensionless Bifurcation Parameter (L is the wave length along the direction of the travelling wave and c is the velocity of the wave), Bifurcation occurs for µ = cos θ. For non-resonant cases, we first give a large family of formal three-dimensional gravity travelling waves, in the form of an expansion in powers of the amplitudes of two basic travelling waves. " Diamond waves " are a particular case of such waves, when they are symmetric with respect to the direction of propagation. The main object of the paper is the proof of existence of such symmetric waves having the above mentioned asymptotic expansion. Due to the occurence of small divisors, the main difficulty is the inversion of the linearized operator at a non trivial point, for applying the Nash Moser theorem. This operator is the sum of a second order differentiation along a certain direction, and an integro-differential operator of first order, both depending periodically of coordinates. It is shown that for almost all angles θ, the 3-dimensional travelling waves bifurcate for a set of " good " values of the Bifurcation Parameter having asymptotically a full measure near the Bifurcation curve in the Parameter plane (θ, µ).

  • Small divisor problem in the theory of three-dimensional water gravity waves
    En attente, 2006
    Co-Authors: Gérard Iooss, Pavel Plotnikov
    Abstract:

    We consider doubly-periodic travelling waves at the surface of an infinitely deep perfect fluid, only subjected to gravity $g$ and resulting from the nonlinear interaction of two simply periodic travelling waves making an angle $2\theta $ between them. \newline Denoting by $\mu =gL/c^{2}$ the dimensionless Bifurcation Parameter ( $L$ is the wave length along the direction of the travelling wave and $c$ is the velocity of the wave), Bifurcation occurs for $\mu =\cos \theta$. For non-resonant cases, we first give a large family of formal three-dimensional gravity travelling waves, in the form of an expansion in powers of the amplitudes of two basic travelling waves. "Diamond waves" are a particular case of such waves, when they are symmetric with respect to the direction of propagation.\newline \emph{The main object of the paper is the proof of existence} of such symmetric waves having the above mentioned asymptotic expansion. Due to the \emph{occurence of small divisors}, the main difficulty is the inversion of the linearized operator at a non trivial point, for applying the Nash Moser theorem. This operator is the sum of a second order differentiation along a certain direction, and an integro-differential operator of first order, both depending periodically of coordinates. It is shown that for almost all angles $\theta $, the 3-dimensional travelling waves bifurcate for a set of "good" values of the Bifurcation Parameter having asymptotically a full measure near the Bifurcation curve in the Parameter plane $(\theta ,\mu ).$

Guanqun Wang - One of the best experts on this subject based on the ideXlab platform.

  • target wave to spiral wave pattern transition in a discrete belousov zhabotinsky reaction driven by inactive resin beads
    Physical Review E, 2010
    Co-Authors: Guanqun Wang, Qingsheng Wang, Srinivasa Rao Pullela, Manuel Marquez, Zhengdong Cheng
    Abstract:

    Wave pattern formation and transition in chemical and biochemical reaction systems can reveal the system properties. We investigate the pattern transition from target waves to spiral waves upon the increment of inactive beads in a discrete system model, where ion-exchange resin loaded with Belousov-Zhabotinsky catalyst corresponds to the active beads. The results show that inactive beads slow the propagation speed of target waves and increase the wave frequency. As the population of inactive beads increases, clusters are formed, which then break waves into segments where bounded spiral pairs are generated and separated into individual spirals. From this observation, we conclude that the population of inactive resin beads acts as the Bifurcation Parameter controlling the wave pattern transition from targets to spirals.

Gérard Iooss - One of the best experts on this subject based on the ideXlab platform.

  • Small divisor problem in the theory of three-dimensional water gravity waves
    Memoirs of the American Mathematical Society, 2009
    Co-Authors: Gérard Iooss, Pavel Plotnikov
    Abstract:

    We consider doubly-periodic travelling waves at the surface of an infinitely deep perfect fluid, only subjected to gravity g and resulting from the nonlinear interaction of two simply periodic travelling waves making an angle 2θ between them. Denoting by µ = gL/c 2 the dimensionless Bifurcation Parameter (L is the wave length along the direction of the travelling wave and c is the velocity of the wave), Bifurcation occurs for µ = cos θ. For non-resonant cases, we first give a large family of formal three-dimensional gravity travelling waves, in the form of an expansion in powers of the amplitudes of two basic travelling waves. " Diamond waves " are a particular case of such waves, when they are symmetric with respect to the direction of propagation. The main object of the paper is the proof of existence of such symmetric waves having the above mentioned asymptotic expansion. Due to the occurence of small divisors, the main difficulty is the inversion of the linearized operator at a non trivial point, for applying the Nash Moser theorem. This operator is the sum of a second order differentiation along a certain direction, and an integro-differential operator of first order, both depending periodically of coordinates. It is shown that for almost all angles θ, the 3-dimensional travelling waves bifurcate for a set of " good " values of the Bifurcation Parameter having asymptotically a full measure near the Bifurcation curve in the Parameter plane (θ, µ).

  • Small divisor problem in the theory of three-dimensional water gravity waves
    En attente, 2006
    Co-Authors: Gérard Iooss, Pavel Plotnikov
    Abstract:

    We consider doubly-periodic travelling waves at the surface of an infinitely deep perfect fluid, only subjected to gravity $g$ and resulting from the nonlinear interaction of two simply periodic travelling waves making an angle $2\theta $ between them. \newline Denoting by $\mu =gL/c^{2}$ the dimensionless Bifurcation Parameter ( $L$ is the wave length along the direction of the travelling wave and $c$ is the velocity of the wave), Bifurcation occurs for $\mu =\cos \theta$. For non-resonant cases, we first give a large family of formal three-dimensional gravity travelling waves, in the form of an expansion in powers of the amplitudes of two basic travelling waves. "Diamond waves" are a particular case of such waves, when they are symmetric with respect to the direction of propagation.\newline \emph{The main object of the paper is the proof of existence} of such symmetric waves having the above mentioned asymptotic expansion. Due to the \emph{occurence of small divisors}, the main difficulty is the inversion of the linearized operator at a non trivial point, for applying the Nash Moser theorem. This operator is the sum of a second order differentiation along a certain direction, and an integro-differential operator of first order, both depending periodically of coordinates. It is shown that for almost all angles $\theta $, the 3-dimensional travelling waves bifurcate for a set of "good" values of the Bifurcation Parameter having asymptotically a full measure near the Bifurcation curve in the Parameter plane $(\theta ,\mu ).$