A cardinal function method of solution of the equation $\Delta u=u-u^{3}$

Abstract:

The steady-state form of the Klein-Gordon equation is given by $(^\ast )$ \[ \Delta u = u - {u^3},\quad u = u(X),\quad X \in {R^3}.\] For solutions which are spherically symmetric, $(^\ast )$ takes the form $\ddot u + 2\dot u/r = u - {u^3}$, $u = u(r)$, where r is the distance from the origin in ${R^3}$. The function $y = ru$ satisfies ${(^\ast }^\ast )$ \[ \ddot y = y - {y^3}/{r^2}.\] It is known that ${(^\ast }^\ast )$ has solutions $\{ {y_n}\} _{n = 0}^\infty$, where ${y_n}$ has exactly n zeros in $(0,\infty )$, and where $y(0) = y(\infty ) = 0$. In this paper, an approximation is obtained for the solution ${y_0}$ by minimizing a certain functional over a class of functions of the form \[ \sum \limits _{k = - m}^m {{a_k}\;} {\operatorname {sinc}}\left [ {\frac {{r - k{h_m}}}{{{h_m}}}} \right ].\] It is shown that the norm of the error is $O({m^{3/8}}\exp ( - \alpha m^{1/2}))$ as $m \to \infty$, where $\alpha$ is positive.