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An algebraic root can depend on a parameter through fractional powers even when the polynomial coefficients have ordinary power series. For example, the roots of $t^3={ϵ}^2$ are on the scale ${ϵ}^{2/3}$. Dominant balance for algebraic roots identifies the exponent, and local expansion of the scaled root supplies successive coefficients.

The small root and the root near $-2$ are regular perturbation roots: they stay finite as the parameter vanishes. Substitution of power series givesFor the third root, dominant balance for algebraic roots requires $\lambda =\mu /\epsilon$. The largest terms give ${\mu }^2(\mu +1)=0$, and the missing branch is $\mu =-1$. The sum of the three roots then yieldsThus the leading roots are $-\epsilon /2$, $-2$, and $-1/\epsilon$; only the last is a divergent perturbation root. The first regular branch happens to have a zero limit, so retaining its first nonzero term is essential on the long spatial scale.

First take $\epsilon \to 0^{+}$, the decaying half-line regime. If the three exact roots are $r_j$, the exact initial value problem solution isThis follows either by solving the three initial-value equations or from the Laplace transform $\hat{y}(s)=\epsilon /(\epsilon s^3+s^2+2s+\epsilon )$. The coefficient formula has sums $\sum A_j=\sum r_j{A}_j=0$ and $\sum r_j^2{A}_j=1$.

Retain the three leading roots but compute their coefficients without expanding their denominators. Set $a=\epsilon /2$, $b=2$, $d=1/\epsilon$. A convenient composite asymptotic expansion isIt satisfies all three initial conditions exactly and retains the fast transient, the ordinary decay, and the slow decay. Its first two coefficients are $\epsilon /2+O({\epsilon }^2)$ and $-\epsilon /2+O({\epsilon }^2)$; the fast coefficient is ${\epsilon }^2+O({\epsilon }^3)$. Consequently the simpler bulk expression is $\frac{\epsilon }{2}(e^{-\epsilon x/2}-e^{-2x})$, but that expression alone does not reproduce the initial derivative layer.

The absolute error estimate isTo see its uniformity, the slow-root error is $O({\epsilon }^2)$, while its magnitude is $O(\epsilon )$ and its coefficient is $O(\epsilon )$. The uniform error bound for nearby decaying exponentials therefore gives an $O({\epsilon }^2)$ contribution even for $x=O({\epsilon }^{-1})$. The middle-root error is $O(\epsilon )$ with coefficient $O(\epsilon )$, again giving $O({\epsilon }^2)$. The fast-root error is $O(1)$ with magnitude $O({\epsilon }^{-1})$ and coefficient $O({\epsilon }^2)$, giving only $O({\epsilon }^3)$. Coefficient errors are $O({\epsilon }^2)$ or smaller. This estimate concerns $y$ itself, rather than asserting the same uniform order for all derivatives or a relative error at its zero.

For the sketch, on $x=\epsilon X$ one obtainsThus $y\sim x^2/2$ at the origin and rises smoothly after the fast layer. For $x=O(1)$ it rises towards a plateau of height approximately $\epsilon /2$, then decays on the much longer scale $x=O({\epsilon }^{-1})$. The bulk maximum lies near $x=\log (4/\epsilon )/(2-\epsilon /2)$ and has height asymptotic to $\epsilon /2$.

The sign of the parameter matters on an infinite interval. If $\epsilon \to 0^{-}$ is allowed, the singular mode grows rather than decays. For each fixed $x>0$, its dominant contribution is $y(x)\sim {\epsilon }^2\exp [(-1/\epsilon +2)x]$. The extra $2$ in that exponent is needed for relative leading accuracy at fixed $x$. The positive-parameter uniform absolute bound and decaying sketch do not extend to that regime.

For $ϵ>0$, dominant balance for algebraic roots gives two roots on the scale $\sqrt{ϵ}$ and a third on the scale ${ϵ}^{2/3}$. On the first scale put $t=\sqrt{ϵ}s$. The equation becomes $s^5-s^3+\sqrt{ϵ}=0$. The derivative of $s^5-s^3$ at either $s=1$ or $s=-1$ is two, so the first correction is $-\sqrt{ϵ}/2$. On the second scale put $t={ϵ}^{2/3}v$; then $1-v^3+{ϵ}^{1/3}{v}^5=0$. Writing $v=1+a{ϵ}^{1/3}+\cdots$ gives $a=1/3$. Thus all three real-root expansions for positive $ϵ$ areThese are fractional-power root expansions. To check that none is missing, the derivative of the polynomial is $t^2(5t^2-3ϵ)$. Its nonzero critical points are $\pm \sqrt{3ϵ/5}$; for sufficiently small positive $ϵ$ the polynomial is positive at the negative one and negative at the positive one. Its three monotone ranges therefore contain exactly these three real roots. The remaining two small roots correspond to the nonreal cube roots of unity in the second balance.

The printed limit does not specify the sign of $ϵ$. If a two-sided real limit is intended, also put $ϵ=-\eta$ with $\eta >0$. The polynomial $t^5+\eta t^3-{\eta }^3$ is strictly increasing and has just one real root. The same dominant balance for algebraic roots gives the negative-parameter branch