Analytical-numerical investigation of bubble-type solutions of nonlinear singular problems

In this work we are concerned about a singular boundary value problem for a second-order nonlinear ordinary differential equation, arising in hydrodynamics and nonlinear field theory, when centrally symmetric bubble-type solutions are sought. We are interested on solutions of this equation which are strictly increasing on the positive semi-axis and have finite limits at zero and infinity. Necessary conditions for the existence of such solutions are obtained in the form of a restriction on the equation parameters. The asymptotic behavior of certain solutions of this equation is analyzed near the two singularities (when r->0+ and r->~), where the considered boundary conditions define one-parameter families of solutions. Based on the analytic study, an efficient numerical method is proposed to compute approximately the needed solutions of the above problem. Some results of the numerical experiments are displayed and their physical interpretation is discussed.

[1]  Boundary value problems for functional differential equations , 1995 .

[2]  W. Rother,et al.  Nonlinear scalar field equations , 1992, Differential and Integral Equations.

[3]  Filippo Gazzola,et al.  Existence of ground states and free boundary problems for quasilinear elliptic operators , 2000, Advances in Differential Equations.

[4]  Pierre-Louis Lions,et al.  Nonlinear scalar field equations, I existence of a ground state , 1983 .

[5]  J. Baxley.  BOUNDARY VALUE PROBLEMS ON INFINITE INTERVALS , 1995 .

[6]  Milton Abramowitz,et al.  Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables , 1964 .

[7]  W. D. Evans,et al.  PARTIAL DIFFERENTIAL EQUATIONS , 1941 .

[8]  Irene A. Stegun,et al.  Handbook of Mathematical Functions. , 1966 .

[9]  A. M. Li︠a︡punov.  Problème général de la stabilité du mouvement , 1949 .

[10]  Stephen Wolfram,et al.  The Mathematica Book , 1996 .

[11]  W. Wasow.  Asymptotic expansions for ordinary differential equations , 1965 .

[12]  Sergey Gavrilyuk,et al.  Media with equations of state that depend on derivatives , 1996 .

[13]  Henri Gouin,et al.  Nucleation of spherical shell-like interfaces by second gradient theory: numerical simulations , 1996, 0906.1897.

[14]  Jan Andres,et al.  Boundary value problems on infinite intervals , 1999 .

[15]  H. Gouin,et al.  An analytical approximation of density profile and surface tension of microscopic bubbles for Van Der Waals fluids , 1997 .

[16]  G. Derrick.  Comments on Nonlinear Wave Equations as Models for Elementary Particles , 1964 .

[17]  N. Konyukhova.  Singular cauchy problems for systems of ordinary differential equations , 1983 .