Method of multiple scales

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.

For $x^{\prime \prime }+3ϵy{x}^{\prime }+x=2(y^{\prime }{)}^2$ and $y^{\prime }=ϵ(1+x-y)$ with leading oscillation $x=A(\tau )\sin t$, $\tau =ϵt$, averaging the auxiliary equation gives $Y^{\prime }=1-Y$. Cancellation of resonant forcing in the oscillator gives $2A^{\prime }+3YA=0$. With $Y(0)=3$ and $A(0)=1$, $Y=1+2e^{-\tau }$ and the displayed amplitude follows. Bounded oscillatory corrections repair the instantaneous auxiliary equation; secular terms in a fixed-time expansion show why the slow scale is needed.

For $y^{\prime \prime }+ϵ(y^{\prime }{)}^n+y=0$, the method of multiple scales gives no first-order phase drift. If $n$ is odd, the amplitude obeys the displayed damping law with $C_n=2^{-(n+1)}\left(\genfrac{}{}{0ex}{}{n+1}{(n+1)/2}\right)$. If $n$ is even, the average is zero and the amplitude is unchanged on the first slow time $T=ϵt$. The odd case dissipates mechanical energy, whereas the even case is reversible and its force is not damping of both velocity signs.

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.

For $x^{\prime \prime }+\epsilon |x|x^{\prime }+x=0$ and unit initial amplitude, averaging uses $⟨|\cos \psi |{\sin }^2\psi ⟩=2/(3\pi )$. Thus $A_T=-2A^2/(3\pi )$ and the displayed algebraic envelope follows. The leading phase has no first-order drift. The dissipation is amplitude dependent, so an exponential envelope with a constant decay rate would give the wrong long-time scale.

For $x^{\prime \prime }+\epsilon (x^2-1)x^{\prime }+x=0$, the averaged amplitude for position-dependent damping has the displayed logistic-type equation. With $A(0)=1$, its positive solution is $A(T)=2/\sqrt{1+3e^{-T}}$. Small amplitudes grow and large ones decay toward the stable value two; the leading oscillation is $A(\epsilon t)\cos t$.

For $x^{\prime \prime }+ϵc(1-x^2)x^{\prime }+x=k$, use $T=ϵt$ and $x=k-r(T)\cos (t+\varphi (T))$. Removal of secular terms gives ${\varphi }_T=0$ and the displayed amplitude equation. With $R=r^2$, $R_T=c(k^2-1)R+(c/4)R^2$, which is explicitly integrable. If $c$ is replaced by a slowly varying coefficient $C(T)$, the same solution uses the accumulated time ${\int }_0^{T}C(s)ds$. The method of multiple scales is uniform on bounded slow-time intervals only while its amplitude and correction estimates remain bounded.

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

For $x^{\prime \prime }+x=-2\epsilon y{x}^{\prime }$ and $y^{\prime }=\epsilon g(x)$, the method of multiple scales gives the displayed amplitude-phase equations, together with constant leading phase. For $g(x)=x$, $Y$ is constant and $R$ changes exponentially. For $g(x)=\log |x|$ and positive amplitude, the integrable phase average is $\log (R/2)$. Thus $L=\log (R/2)$ obeys $L_{TT}+L=0$ with $L_T=-Y$. The identically zero oscillator is inadmissible for logarithmic feedback, whose force is undefined there.

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 initial amplitude one, the method of multiple scales gives $x\simeq A(ϵt)\sin t$ with $A_T=-4A^2/(3\pi )$, hence $A=1/(1+4ϵt/(3\pi ))$. Positive damping causes algebraic decay. Negative damping yields a formal finite-time amplitude pole, but the weak-damping assumption fails before it because $|ϵ|A$ becomes order one. A leading approximation has order-$|ϵ|$ displacement error on fixed slow-time intervals away from that pole.

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.

Three Fourier modes form a spatial resonance when their wavevectors sum to zero. Products of the conjugate second and third modes then force the first mode, with cyclic counterparts. Such terms couple both amplitudes and phases, unlike self-saturation terms proportional only to squared magnitudes. For three equal wavenumbers, the vectors are separated by $120$ degrees and support hexagonal pattern interactions.

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 $y^{\prime \prime }+({\omega }^2+ϵ\cos t)y=0$, use the method of multiple scales with $T=ϵt$ and $y_0=a(T)\cos (t/2)+b(T)\sin (t/2)$. Removing secular terms gives $a_T=(k-1/2)b$ and $b_T=-(k+1/2)a$. The slow eigenvalues satisfy ${\lambda }^2=1/4-k^2$, so an exponentially growing mode exists inside the displayed leading instability interval. At its leading endpoints the slow matrix is nilpotent; higher-order corrections are needed to determine the exact edges. Floquet theory distinguishes this exponential instability from algebraic growth at a repeated multiplier.

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 $x^{\prime \prime }+[1+ϵ\cos t+f{ϵ}^2]x=0$, let $T={ϵ}^{2}t$ and $x_0=a(T)\cos t+b(T)\sin t$. First-order forcing produces a constant and second harmonics, not a resonant harmonic. Their feedback at second order gives the displayed solvability condition in the method of multiple scales. The slow squared growth rate is $(f+1/12)(5/12-f)/4$. Consequently the leading instability interval is $-1/12<f<5/12$; outside its closure the slow amplitudes oscillate. The endpoints are degenerate and require the fourth-order edges of the second Mathieu instability tongue for exact small-parameter stability.

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.