Nonlinear Stokes phenomena in first or second order differential equations

Details Nonlinear Stokes phenomena in first or second order differential equations Nonlinear Stokes phenomena in first or second order differential equations

2006-08-12

arxiv.org/abs/math/0608303v1

Mathematics

Classical Analysis and ODEs

Scientific paper

We study singularity formation in nonlinear differential equations of order $m\leqslant 2$, $y^{(m)}=A(x^{-1},y)$. We assume $A$ is analytic at $(0,0)$ and $\partial_y A(0,0)=\lambda\ne 0$ (say, $\lambda=(-1)^m$). If $m=1$ we assume $A(0,\cdot)$ is meromorphic and nonlinear. If $m=2$, we assume $A(0,\cdot)$ is analytic except for isolated singularities, and also that $\int_{s_0}^\infty |\Phi(s)|^{-1/2}d|s|<\infty$ along some path avoiding the zeros and singularities of $\Phi$, where $\Phi(s)=\int_{0}^s A(0,\tau)d\tau$. Let $H_{\alpha}=\{z:|z|>a>0,\arg(z)\in (-\alpha,\alpha)\}$. If the Stokes constant $S^+$ associated to $\RR^+$ is nonzero, we show that all $y$ such that $\lim_{x\to +\infty}y(x)=0$ are singular at $2\pi i$-quasiperiodic arrays of points near $i\RR^+$. The array location determines and is determined by $S^+$. Such settings include the Painlev\'e equations $P_I$ and $P_{II}$. If $S^+=0$, then there is exactly one solution $y_0$ without singularities in $H_{2\pi-\epsilon}$, and $y_0$ is entire iff $y_0=A(z,0)\equiv 0$. The singularities of $y(x)$ mirror the singularities of the Borel transform of its asymptotic expansion, $\mathcal{B}\tilde{y}$, a nonlinear analog of Stokes phenomena. If $m=1$ and $A$ is a nonlinear polynomial with $A(z,0)\not\equiv 0$ a similar conclusion holds even if $A(0,\cdot)$ is linear. This follows from the property that if $f$ is superexponentially small along $\RR^+$ and analytic in $H_{\pi}$, then $f$ is superexponentially unbounded in $H_{\pi}$, a consequence of decay estimates of Laplace transforms.

Affiliated with

Also associated with

No associations

Rating

Nonlinear Stokes phenomena in first or second order differential equations does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with Nonlinear Stokes phenomena in first or second order differential equations, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Nonlinear Stokes phenomena in first or second order differential equations will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-251839