For the Cauchy problem d2w/dt2+N(dw/dt,w,t)=f(t)d^2w/dt^2+N(dw/dt,w,t)=f(t), Frasca and Khurshudyan (2018) proposed a nonlinear Green's function (NLG) method: the nonlinear Green's function GG solves the same equation with a Dirac delta source, and the solution of the forced equation is represented by a 'short time' convolution expansion whose first term w(t)≈a0∫0tG(t−τ)f(τ)dτw(t)\approx a_0\int_0^t G(t-\tau)f(\tau)d\tau is used in practice. The representation G(t)=θ(t)w0(t)G(t)=\theta(t)w_0(t), with θ\theta the Heaviside function and w0w_0 the solution of the homogeneous Cauchy problem, was proved under a multiplicativity condition on NN. Because θ\theta takes only the values 0 and 1, this condition holds for every quasi-linear nonlinearity, so what remains unsupported is the complete absence of a rigorous theory behind the approximation itself: no convergence result, no a priori error bounds, no domain of validity, no truncation criteria, no principled choice of the free scale parameters a0a_0 and ss (currently tuned numerically), and no analysis of when the spectral frequencies read off from the leading-order GG are reliable. The problem is to establish such a theory, or to refute the method by exhibiting equations in the covered class for which the expansion fails.

Background

For a linear ordinary differential equation, the solution of the inhomogeneous Cauchy problem is given by convolving the source with the Green's function — the solution of the same equation driven by a Dirac delta distribution δ\delta. Superposition fails for nonlinear equations, so no exact counterpart exists in general. Frasca (Mod. Phys. Lett. A 22, 2007; Int. J. Mod. Phys. A 23, 2008) introduced a heuristic extension, the nonlinear Green's function (NLG) method: for d2w/dt2+N(w,t)=f(t)d^2w/dt^2+N(w,t)=f(t), let GG solve d2G/dt2+N(G,t)=sδ(t)d^2G/dt^2+N(G,t)=s\delta(t) with Cauchy data G(0)=0G(0)=0, G′(0)=sG'(0)=s, where ss is a free real scale parameter, and postulate the 'short time expansion' w(t)=a0∫0tG(t−τ)f(τ)dτ+∑k≥1ak∫0t(t−τ)kG(t−τ)f(τ)dτw(t)=a_0\int_0^t G(t-\tau)f(\tau)d\tau+\sum_{k\ge 1}a_k\int_0^t(t-\tau)^kG(t-\tau)f(\tau)d\tau with further coefficients aka_k fixed by the initial data w(k)(0)w^{(k)}(0) and a further free parameter a0a_0. Only the first term is used in applications; its accuracy has only ever been assessed empirically, by comparison with a reference numerical solution computed by the method of lines (a standard technique that discretizes space and integrates the resulting ODE system in time). Frasca and Khurshudyan (arXiv:1805.10495; Comput. Math. Methods 2019) proved a representation theorem for nonlinearities N(w,t)N(w,t): if the multiplicativity condition N(θw,t)=θ(t)N(w,t)N(\theta w,t)=\theta(t)N(w,t) holds, where θ\theta is the Heaviside step function, then G(t)=θ(t)w0(t)G(t)=\theta(t)w_0(t), where w0w_0 solves the homogeneous equation with w0(0)=0w_0(0)=0, w0′(0)=sw_0'(0)=s. The source paper (Frasca and Khurshudyan, arXiv:1806.00274, a preprint not published in a journal) extends this to equations nonlinear in the first derivative, d2w/dt2+N(dw/dt,w,t)=f(t)d^2w/dt^2+N(dw/dt,w,t)=f(t): under the analogous condition N(θ dw/dt,θw,t)=θ(t)N(dw/dt,w,t)N(\theta\,dw/dt,\theta w,t)=\theta(t)N(dw/dt,w,t), the same representation G(t)=θ(t)w0(t)G(t)=\theta(t)w_0(t) holds (Theorem 2), and the paper derives explicit GG for traveling-wave reductions of the generalized Burgers equation, a nonlinear heat equation with power temperature gradient, and a wave equation with nonlinear damping, reporting tuned logarithmic errors of order 10−310^{-3} to 10−810^{-8} against method-of-lines solutions. One structural fact, made explicit only later, frames the problem correctly: since θ\theta takes only the values 0 and 1, any quasi-linear nonlinearity — any sum of monomials in (dw/dt,w)(dw/dt,w) of joint degree at least 1, which covers every example in the literature — satisfies the multiplicativity condition automatically (Khurshudyan, Comput. Appl. Math. 39, 2020, proved the underlying identity θp=θ\theta^p=\theta almost everywhere and applied it to the forced cubic Duffing and fractionally damped Liénard equations). So the representation theorem itself is nearly unconstrained; what remains completely unsupported is the analytic status of the short-time expansion built on top of it, and the related practice, also practiced in the source paper, of reading off the 'spectrum' of the nonlinear operator from the Fourier content of the leading-order GG under the unanalyzed assumption that all corrections are higher order.

Problem Statement

Establish a rigorous approximation theory — or a refutation — of the nonlinear Green's function (NLG) short-time expansion for second-order Cauchy problems nonlinear in the first derivative, $$ \frac{d^2w}{dt^2}+N!\left(\frac{dw}{dt},w,t\right)=f(t),\qquad t>0,\qquad w(0)=u_0,\quad w'(0)=v_0,

with the nonlinearity $N$ in the class covered by the representation theorem of Frasca–Khurshudyan (arXiv:1806.00274, Theorem 2) — concretely, nonlinearities satisfying the multiplicativity condition $N(\theta\,dw/dt,\theta w,t)=\theta(t)N(dw/dt,w,t)$ with $\theta$ the Heaviside step function, a condition automatically satisfied by every quasi-linear $N$ (every finite sum of monomials in $(dw/dt,w)$ of joint degree at least 1, with or without $t$-dependent coefficients), which comprises all examples treated in the literature — for which the nonlinear Green's function is $G(t)=\theta(t)w_0(t)$ with $w_0$ solving the homogeneous equation with $w_0(0)=0$, $w_0'(0)=s$. Establish a theory that settles the following: (i) Prove or disprove that the first-term approximation $w(t)\approx a_0\int_0^t G(t-\tau)f(\tau)\,d\tau$ has an a priori error bound, explicit in the problem data $f$, $N$, the Cauchy data $u_0$ and $v_0$, the scale parameters $a_0$ and $s$, and the time horizon, and determine the domain of validity in time and amplitude. (ii) Settle the convergence (or divergence/asymptotic-only character) of the full short-time expansion $w(t)=\sum_{k\ge 0}a_k\int_0^t (t-\tau)^kG(t-\tau)f(\tau)\,d\tau$. (iii) Give criteria for when higher-order terms of that expansion must be retained. (iv) Give a principled rule for selecting $a_0$ and $s$ from the problem data. (v) Give a quantified effect of that choice of $a_0$ and $s$ on the error. (vi) Give conditions under which spectral frequencies extracted from the Fourier content of the leading-order $G$ (as practiced in the source paper for the sine-Gordon and $\phi^4$ models, i.e. the nonlinear Klein–Gordon equations with a sine and a quartic potential respectively) are reliable, i.e. conditions under which the neglected higher-order corrections provably do not shift the leading frequencies. Alternatively, the method is refuted by exhibiting an explicit equation within the covered class, with a specified source $f$ and Cauchy data, and a finite time interval $[0,T]$ on which the method claims validity, such that for every real choice of the scale parameters $a_0$ and $s$ with $a_0\neq 0$ and $s\neq 0$ (excluding the degenerate choices for which the first-term approximation vanishes identically) at least one of the following two independently checkable failures occurs: (a) the error of the first-term NLG approximation on $[0,T]$ fails to be $O(\varepsilon)$ as the amplitude parameter $\varepsilon$ of the source $f$ and of the Cauchy data tends to zero; or (b) the full short-time expansion fails on $[0,T]$ to converge, or converges to a limit other than the solution, or is not asymptotic to the solution in the sense of (ii). ## References 1. 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