Singular perturbation

A singular perturbation multiplies a highest derivative or otherwise changes the limiting equation's order when its small parameter is set to zero. It commonly separates fast and slow time scales.

Adiabatic elimination replaces a rapidly relaxing variable by its instantaneous stable quasistatic value. If $\dot{R}=-\Gamma (t)R+P(t)$ with $\Gamma >0$, write $R=R_0+\delta R$, $R_0=P/\Gamma$. Then $\dot{\delta R}+\Gamma \delta R=-{\dot{R}}_0$. After the transient, $\delta R\simeq -{\dot{R}}_0/\Gamma$ when coefficients vary slowly compared with ${\Gamma }^{-1}$. This gives both the leading reduction and its time-scale error; rapid relaxation alone does not justify an additional Taylor expansion in another parameter.

A matched asymptotic expansion constructs approximations in regions with different distinguished scalings and determines their free constants by requiring agreement in an overlap region.

The equation $ϵy^{\prime \prime }+x{y}^{\prime }+y=0$ integrates to $ϵy^{\prime }+xy=C$. Equal endpoint data $y(-1)=y(1)=A$ force $C=0$ and give the unique Gaussian solution above. Its central width is $\sqrt{ϵ}$, with amplitude $A{e}^{1/(2ϵ)}$. The exponentially large central mode is missed by an assumed bounded algebraic outer expansion $C/x$. Changes of order one relative to either endpoint value occur over distance $O(ϵ)$ from that endpoint.

For $ϵy^{\prime \prime }+s^2{y}^{\prime }+y=0$ on $s\ge 0$, removing the drift by $y=e^{-s^3/(6ϵ)}v$ givesThe transformed potential changes sign at $s_t\sim \sqrt{2}{ϵ}^{1/4}$. WKB approximation is oscillatory inside and exponential outside this turning region; an Airy turning-point connection formula applies over width ${ϵ}^{5/12}$. At the endpoint the wavelength is $\sqrt{ϵ}$, while the physical exponential prefactor changes on the envelope scale ${ϵ}^{1/3}$. Balancing only diffusion and drift and declaring a single ${ϵ}^{1/3}$ layer misses the zeroth-order term and the turning region. Matching constants may be singular at homogeneous boundary-value resonances.

For $ϵy^{\prime \prime }+a(x)y^{\prime }+b(x)y=0$ with smooth $a$ bounded positively away from zero, the first-order equation for the outer expansion is selected by the right boundary condition. The left mismatch is repaired on the scale $\xi =x/ϵ$. If $a(0)=a_0>0$, the leading inner transient is $B{e}^{-a_0\xi }$. Its next equation is $Y_1^{\prime \prime }+a_0{Y}_1^{\prime }=-a^{\prime }(0)\xi Y_0^{\prime }-b(0)Y_0$. Matching its constant and linear large-$\xi$ terms to the outer expansion permits an order-$ϵ$ composite approximation. A sign reversal of the drift places the transient at the opposite endpoint.

If a leading outer expansion diverges as $\log x$ and a derivative coefficient contains $x-\epsilon z$, balancing that nonlinear correction gives a layer larger than $O(\epsilon )$. Write $x=\epsilon LX$ with $L=\log (1/\epsilon )$ and subtract the large negative constant from $z$. Matching often introduces a $\log L$ correction and yields an endpoint slope of order $1/(\epsilon L)$. The balance must be derived from the actual equation; logarithmic divergence alone does not fix a universal layer width.

For $\epsilon x{w}^{\prime \prime }+w^{\prime }+2xw=0$ with left endpoint $x=\epsilon$, diffusion there is of size ${\epsilon }^2$. The distinguished coordinate is $z=(x-\epsilon )/{\epsilon }^2$, giving $(1+\epsilon z)W^{\prime \prime }+W^{\prime }+2{\epsilon }^3(1+\epsilon z)W=0$. Matching to one with zero left value gives $W=1-e^{-z}-\epsilon z(z+2)e^{-z}/2+O({\epsilon }^2)$. The layer width is different from the endpoint's distance from zero.

In $\epsilon y^{\prime \prime }+2x{y}^{\prime }-2xy=0$, the drift vanishes at the endpoint $x=0$. Balancing diffusion against drift gives the inner expansion coordinate $z=x/\sqrt{\epsilon }$. The leading layer equation is $Y^{\prime \prime }+2z{Y}^{\prime }=0$, with a bounded error function profile. A regular outer first-order calculation alone cannot impose both endpoint values.

If an outer expansion contains $-(\epsilon /2)\log x$ and the layer coordinate is $x=\sqrt{\epsilon }z$, its overlap contains $-(\epsilon /2)\log z-(\epsilon /4)\log \epsilon$. Matching forces an inner expansion term $-(\epsilon /4)\log \epsilon \mathrm{erf}z$ when that homogeneous profile tends to one in the overlap and vanishes at the boundary. The logarithm multiplies the matching profile, rather than an arbitrary constant.

An overlap region is a parameter-dependent range in which two neighboring asymptotic expansions have simultaneously ordered common limits. Between scales ${ϵ}^2$ and $ϵ$, it is ${ϵ}^2\ll x\ll ϵ$: the narrow coordinate tends to infinity while the wider coordinate tends to zero. Matching uses these limiting descriptions, not an assertion that either expansion is uniform at all values of the other's scaled variable.

An intermediate asymptotic region has a scale between two other scales and provides their connecting matched asymptotic expansion. It can be needed because a coefficient changes its dominant form before the narrowest region's highest-derivative balance is reached. For example, a coefficient $x+ϵ$ changes near $x=O(ϵ)$ even when the fast inner expansion has width $O({ϵ}^2)$. The intermediate solution must be matched at both ends rather than assigned an independent boundary value.

In ${\epsilon }^2{y}^{\prime \prime }+a(x)y^{\prime }+\cdots =f$, a simple zero $a(x_{\ast })=0$, $a^{\prime }(x_{\ast })\ne 0$, balances diffusion and advection on width $\epsilon$. With $a(x)\simeq 2(x-x_{\ast })$, the leading homogeneous inner equation is $Y^{\prime \prime }+2z{Y}^{\prime }=0$, whose bounded solutions are constants and the error function. Thus distinct order-one outer solutions can match across an interior transition. Reversing the diffusion sign makes the nonconstant homogeneous solution grow at both ends, changing the matching structure.

For $-{\epsilon }^2{y}^{\prime \prime }+2(x-x_{\ast })y^{\prime }+\cdots =f$ on an interval straddling $x_{\ast }$, the bounded leading interior solution cannot connect unequal constants. A common bulk value is instead selected by an inner solvability condition, and width-${\epsilon }^2$ endpoint layers enforce the two boundary values. At endpoints $x=0,2$ with $x_{\ast }=1$, their leading decaying factors are $e^{-2x/{\epsilon }^2}$ and $e^{-2(2-x)/{\epsilon }^2}$. The interior layer still corrects derivative mismatches at smaller amplitude.

A distinguished limit rescales variables so that two or more effects enter the leading asymptotic balance together. It identifies a region whose governing equation differs from the equations in adjacent scales.

An inner expansion resolves a region in which a stretched independent variable makes otherwise small effects enter the leading balance.

The singular perturbation equation $ϵ\sqrt{x}{u}^{\prime \prime }+u^{\prime }=0$ balances its derivatives on the width $x={ϵ}^{2}X$. The inner equation is $\sqrt{X}{U}^{\prime \prime }+U^{\prime }=0$, so $U^{\prime }=C{e}^{-2\sqrt{X}}$. Its solution with $U(0)=0$ and $U(\infty )=1$ is the displayed $H$, since ${\int }_0^{\infty }{e}^{-2\sqrt{q}}dq=1/2$. Near zero $H=2X-8X^{3/2}/3+O(X^2)$, so the first derivative is finite while the second is singular. Treat the differential equation on the open interval and impose the endpoint value by continuous extension.

An inner variable is a stretched coordinate such as $\xi =(x-x_0)/\delta (ϵ)$ that remains order one inside a thin region of width $\delta (ϵ)$.

An outer expansion describes the solution away from a thin or distant inner region and is matched to the inner expansion in their common limit.

An additive composite expansion combines inner and outer approximations by adding them and subtracting their common overlap. It is uniformly valid across both regions to the retained order.

When inner and outer approximations share a nonzero multiplicative overlap, a multiplicative composite divides their product by that common part. It is especially convenient when different regions contribute separate exponential or boundary-layer factors.

For $ϵy^{\prime \prime }+a(x)y^{\prime }+b(x)y=0$ with $a>0$, the reduced first-order outer problem takes its boundary condition at the right endpoint. A rapidly decaying exponential in $x/ϵ$ supplies the otherwise unsatisfied left-end condition, so the layer lies at the outflow end for the fast mode associated with this sign convention.

A switchback term such as $ϵ\log ϵ$ is forced into a matched expansion when logarithms of a stretched coordinate split into parameter-dependent and local pieces in the overlap region.

If a stretched variable is $x=ϵr$ and an outer expansion contains ${ϵ}^2\log x$, its expression in the fixed-$r$ region is ${ϵ}^2\log r+{ϵ}^2\log ϵ$. The parameter logarithm must therefore be present in the inner expansion, often multiplying a homogeneous solution selected by a boundary condition. An ansatz containing only integer powers with parameter-independent coefficients misses this switchback term. The same reasoning applies to other powers of $ϵ$ and other stretched-coordinate scalings.

The method of multiple scales treats fast and slow variables such as $t$ and $T=ϵt$ as independent, then removes resonant forcing to obtain slow amplitude and phase equations.

Two fundamental harmonics of frequency one multiply into only a constant and second harmonics. The first-order solvability condition in the method of multiple scales therefore has no amplitude-changing resonant coupling. Their first-order particular corrections can mix back into the fundamental at second order, selecting the possible scale $t=O({ϵ}^{-2})$. Detuning may suppress the resulting resonance, so identifying this scale alone does not prove instability for every initial state.

For $x\prime \prime +ϵf(x)x\prime +x=0$, write $x_0=A(T)\cos (t+\theta )$ and $T=ϵt$. Removing secular terms gives $A_T=-A⟨f(A\cos \psi ){\sin }^2\psi ⟩$ and ${\theta }_T=0$, where the brackets mean a period average. This is the first-order solvability condition in the method of multiple scales, with a slowly evolving amplitude and no first-order frequency correction.

If $f$ is odd, its average $⟨f(A\cos \psi ){\sin }^2\psi ⟩$ vanishes under a half-period shift. Hence the averaged amplitude for position-dependent damping is constant at first order. Odd-power damping creates only even harmonics when multiplied by the leading oscillator velocity; these do not resonate with the fundamental harmonic. Higher-order drift is still possible.

A secular term grows in a nominal perturbation correction and eventually invalidates an otherwise ordered expansion. In the method of multiple scales, a solvability condition removes resonant forcing that would generate such growth.

Whitham modulation theory averages a variational wave equation over a fast periodic phase and derives slow equations for its local wavenumber, frequency, and amplitude.

The averaged Lagrangian is the phase average of a Lagrangian evaluated on a periodic wave profile. Variation with respect to profile parameters gives the local dispersion relation, while variation of the slow phase gives a modulation conservation law.

Multiplying a one-dimensional Euler--Lagrange wave equation by the derivative of its periodic profile with respect to fast phase produces a phase first integral plus slow divergence terms. Phase averaging removes the periodic derivative and yields the slow phase equation.

For local wavenumber $k={\Theta }_X$, frequency $\omega =-{\Theta }_T$, and averaged Lagrangian $\overline{L}(k,\omega ,\dots )$, slow-phase variation gives

For an averaged Lagrangian, the wave-action density is $W=\partial \overline{L}/\partial \omega$, up to the sign convention used for the fast phase.

In a slowly varying conservative medium, wave action obeys a continuity equation with flux equal to wave-action density times group velocity.

For $u^{\prime \prime }+u=ϵf(u,u^{\prime },ϵt)$, write $u_0=R(T)\cos (t+\varphi (T))$ with $T=ϵt$. Removing resonant sine and cosine forcing gives

When the weak force depends only on velocity, the phase average is a periodic total derivative and vanishes. The leading phase is constant, while the amplitude evolves according to the remaining average. The sign of that drift determines damping or growth and must be checked from the force, rather than inferred from its velocity dependence alone.

For $y^{\prime \prime }+y=\epsilon (y^{\prime }{)}^3$ with positive $\epsilon$, the oscillator energy increases at rate $\epsilon (y^{\prime }{)}^4$. The amplitude-phase equations give $R(T)=R_0(1-3R_0^{2}T/4{)}^{-1/2}$. The weakly nonlinear expansion fails when the growing amplitude makes $\epsilon R^2$ large, so its formal finite-time divergence does not itself control the exact late-time trajectory.

For a weak force depending only on position, the amplitude average is a periodic total derivative and vanishes. The leading harmonic amplitude is constant, while the phase may drift. The method of multiple scales agrees with the exact conservation of energy for bounded orbits in the effective potential. A cubic position force on the right side of the oscillator equation lowers the frequency at first order when its coefficient is positive.

A slow time resolves evolution occurring over many periods of a faster process. It is treated as an independent variable in the method of multiple scales.

A weakly nonlinear expansion writes a small disturbance as powers of an amplitude parameter while resolving its evolution on a slow time. Near a critical eigenvalue, linear detuning and nonlinear interactions enter the same perturbation order. The Fredholm solvability condition for a self-adjoint operator, or its adjoint form for a general operator, yields an amplitude equation without requiring the full higher-order correction.

A quadratic product of plane waves generates sums and differences of their wavenumbers and angular frequencies. A component drives a secular term when its generated pair also satisfies the linear dispersion relation. A solvability condition in the method of multiple scales then determines the coupled amplitude equations on a slow time.

Detuning measures the small mismatch from a resonance condition. When the mismatch and nonlinear forcing are both $O(ϵ)$, they enter the same slow time equation. A small shift of the wavenumber can therefore balance a quadratic interaction and permit a locked periodic travelling wave.

A resonant interaction requires matching both wavenumber and angular frequency: $k_3=k_1+k_2$ and ${\omega }_3={\omega }_1+{\omega }_2$ for a sum interaction, with corresponding sign changes for differences. A common phase velocity alone does not guarantee that a generated harmonic lies on the dispersion relation.

A fundamental wave and its second harmonic satisfy phase matching for a quadratic wave interaction when $\omega (2k)=2\omega (k)$. The square of the fundamental drives the second harmonic, while its product with the conjugate second harmonic drives the fundamental. Projecting these terms onto the resonant modes gives coupled amplitude equations rather than a uniformly valid correction at fixed amplitude.

For $A_T=-\overline{A}B/2$ and $B_T=-A^2/4$, write $A=R{e}^{i\theta }$, $B=P{e}^{i\varphi }$ and $\delta =\varphi -2\theta$. The polar equations areWhere the polar phases are defined, direct differentiation gives the first integrals $R^2-2P^2$ and $R^{2}P\sin \delta$. In particular $R^2{\theta }_T$ and $P^2{\varphi }_T$ are constant. If $\delta =\pi$ and $R^2-2P^2=C>0$, then $P_T=P^2/2+C/4$, a Riccati equation whose positive solution develops a finite-time pole. This is finite-time blowup of the reduced amplitude system; it does not establish blowup of the full wave equation beyond the domain of the weakly nonlinear expansion.

Harmonic balance substitutes a finite Fourier series into a differential equation and projects the residual onto the retained harmonics. It determines their amplitudes and phases. A nonlinear equation generally generates additional harmonics, so vanishing of the projected residual alone does not make the finite series an exact solution. In a weakly nonlinear expansion, nonresonant generated harmonics can be found at the next order by solving their linear equations.

At each perturbation order, the coefficient of a homogeneous oscillatory mode must vanish from the forcing to prevent secular growth. This solvability condition determines the slow evolution of lower-order amplitudes.

Parametric resonance occurs when periodic variation of a system parameter couples oscillatory modes and produces exponential amplitude growth. Multiple-scale solvability identifies the instability tongues and their detuning widths.

For weak quadratic coupling between frequencies one and two, the first oscillator's complex amplitude can obey $2i{A}_T+\alpha A+b\overline{A}=0$ with a constant real second-oscillator amplitude $b$. In real coordinates this is a linear slow system with squared growth rate $(b^2-{\alpha }^2)/4$. Exponential amplification occurs when the modulation amplitude exceeds detuning, $|b|>|\alpha |$. The method of multiple scales is valid over bounded slow time while the physical amplitudes remain in the weak-coupling regime. At equality the slow matrix can have a Jordan block and produce linear growth for some initial phases.

For $u^{\prime \prime }+[1+ϵ\sin t+{ϵ}^{2}f]u=0$, the second parametric-instability tongue is $-1/12<f<5/12$ to leading order. Outside its closure every leading slow amplitude is oscillatory.

For $u^{\prime \prime }+[1+k{ϵ}^2+ϵ\cos t]u=0$, set $t=2s$. The Mathieu equation has $a=4+4k{ϵ}^2$ and $q=-2ϵ$. Near the value $4$, the orthonormal even Fourier series basis $1,\cos 2s,\cos 4s,\cos 6s$ gives diagonal entries $0,4,16,36$ and adjacent couplings $\sqrt{2}q,q,q$. The odd basis $\sin 2s,\sin 4s,\sin 6s$ gives diagonal entries $4,16,36$ and adjacent couplings $q,q$. Higher modes require at least six coupling steps to affect the root near $4$, so these truncations determine its coefficients through $q^4$.

Substituting $a=4+c_2{q}^2+c_4{q}^4$ into the two characteristic polynomials and equating coefficients yieldsConverting back gives the displayed gap edges with remainders $O({ϵ}^4)$ in $k$. In a fixed bounded neighborhood of this parametric resonance, the true unstable gap is $k_{-}<k<k_{+}$. The leading slow-amplitude calculation cannot decide exact stability at $k=-1/12$ or $5/12$; the fourth-order shifts put those fixed values just outside the gap for sufficiently small positive $ϵ$. This does not provide a bound uniform in $ϵ$ for their transient amplification.