What is the distributional Green's function of the second-order nonlinear ODE $w''+N(w)=f(t)$ for the transcendental nonlinearities $\cos$, $\cosh$, $\cot$, $\coth$, $\arccos$, $\mathrm{arccot}$, $\mathrm{arccosh}$, $\mathrm{arccoth}$ that fail the generalized multiplicativity property?

For the nonlinear ODE $w''+N(w)=f(t)$, the nonlinear Green's function $G$ is a distributional solution of $G''+N(G)=s\,\delta(t)$. Frasca and Khurshudyan (2019) proved that whenever the nonlinearity satisfies the generalized multiplicativity property $N(\theta\cdot w)=\theta(t)\cdot N(w)$ ($\theta$ the Heaviside step function), $G$ admits the explicit representation $G=\theta\cdot w_0$ with $w_0$ the solution of the homogeneous equation, and they verified the property for powers $w^n$ ($n$ integer), $\sinh$, $\tanh$, $\sin$, $\tan$, $\arcsin$, $\arctan$, $\mathrm{arcsinh}$, $\mathrm{arctanh}$, $\ln(1+w)$ and combinations, while showing that the Liouville case $N(w)=\exp w$ fails it and yields a Green's function with two one-sided contributions. Of their two open problems, the extension of the power case $N(w)=w^n$ from integer to real $n>0$ was subsequently settled at the almost-everywhere/distributional level by Khurshudyan (2020), who proved the real-power Heaviside identity $\theta^{r}=\theta$ a.e. ($r>0$) with a distributional counterpart and applied it to fractionally nonlinear equations. What remains open is the systematic treatment of the transcendental nonlinearities: determining, within a precise theory of nonlinear distributions, the structure of the distributional Green's functions of $\cos$, $\cosh$, $\cot$, $\coth$, $\arccos$, $\mathrm{arccot}$, $\mathrm{arccosh}$, $\mathrm{arccoth}$ and their combinations, which the authors observed to behave like the Liouville case — the further work explicitly announced in the published article.

Background

For a linear non-homogeneous differential equation $Lw=f$, the Green's function method solves $LG=\delta$ (with $\delta$ the Dirac distribution) and represents the solution as the convolution of $G$ with $f$. Because superposition fails, this does not directly extend to nonlinear equations. Nevertheless, Frasca (2007, 2008) introduced a nonlinear analogue: for the second-order nonlinear ODE

$\frac{d^2 w}{dt^2}+N(w)=f(t),\qquad t>0,$

one studies the distributional solution $G$ of

$\frac{d^2 G}{dt^2}+N(G)=s\,\delta(t),$

where $N$ is the nonlinear term and $s$ a scaling constant, and showed that the general solution of the forced equation admits a short-time expansion in terms of $G$: $w(t)=\sum_{k\ge 0}\alpha_k\int_0^t (t-\tau)^k\,G(t-\tau)\,f(\tau)\,d\tau$ with coefficients $\alpha_k$ fixed by the initial data, whose first term is often an accurate approximation. Since $\delta$ is a distribution, $G$ is a distribution and the composition $N(G)$ is not defined in classical distribution theory (products of distributions are generally undefined); algebras of generalized functions such as those of Colombeau, Biagioni and Egorov provide frameworks where such nonlinear operations make sense.

Frasca and Khurshudyan (2019) identified a large class where $G$ is explicit. Let $\theta$ denote the Heaviside step function ($\theta(t)=1$ for $t>0$, $\theta(t)=0$ for $t\le 0$). If $N$ satisfies the generalized multiplicativity property

$N(\theta\cdot w)=\theta(t)\cdot N(w),$

then the nonlinear Green's function admits the representation $G(t)=\theta(t)\,w_0(t)$, where $w_0$ is the general solution of the homogeneous equation $w_0''+N(w_0)=0$ subject to suitable Cauchy data — so the Green's function is obtained simply by switching on the homogeneous solution. The authors verified the property for $N(w)=w^n$ with integer $n$, $\sinh w$, $\tanh w$, $\sin w$, $\tan w$, $\arcsin w$, $\arctan w$, $\mathrm{arcsinh}\,w$, $\mathrm{arctanh}\,w$, $\ln(1+w)$, and various products and powers of these, yielding hierarchies of exactly solvable nonlinear ODEs and, via traveling-wave reduction, PDEs. The proof for $N(w)=w^n$ rests on the identity $\theta^n=\theta$ for integer $n$.

Two exceptions drive the open questions. For the Liouville nonlinearity $N(w)=\exp w$, the multiplicativity property fails since $\exp(\theta w)=1+\theta\,\exp(w)\ne\theta\,\exp(w)$, and the Green's function instead combines two one-sided contributions, $G(t)=2\theta(t)\ln\frac{\sqrt{2\varepsilon}}{\cosh(\sqrt{\varepsilon}t+\phi)}+2\theta(-t)\ln\frac{\sqrt{2\varepsilon}}{\cosh(\sqrt{\varepsilon}t-\phi)}$ for constants $\varepsilon,\phi$. The authors posed two open problems: (i) prove the multiplicativity property for $N(w)=w^n$ with real exponent $n>0$ (proved only for integer $n$, while numerics suggested it holds for all real $n$); (ii) the Liouville-type obstruction was observed numerically to 'seem to hold' also for $\cos$, $\cosh$, $\cot$, $\coth$, $\arccos$, $\mathrm{arccot}$, $\mathrm{arccosh}$, $\mathrm{arccoth}$, and a systematic treatment of these functions was declared to need further work. Problem (i) has since been addressed: Khurshudyan (2020) proved that any positive real power of the Heaviside function equals $\theta$ almost everywhere, established a corresponding identity for the associated distributions, and applied it to construct nonlinear Green's functions for quasi-linear ODEs with fractional (non-integer) power nonlinearities. The transcendental families of problem (ii) — functions that either do not vanish at the origin ($\cos$, $\cosh$), are singular there ($\cot$, $\coth$), or have restricted real domains ($\arccos$, $\mathrm{arccot}$, $\mathrm{arccosh}$, $\mathrm{arccoth}$) — remain untreated, and it is this surviving core that constitutes the present problem.

Problem Statement

For the second-order nonlinear ODE

$\frac{d^2 w}{dt^2}+N(w)=f(t),\qquad t>0,$

let the nonlinear Green's function $G$ be a distributional solution of

$\frac{d^2 G}{dt^2}+N(G)=s\,\delta(t),$

with $\delta$ the Dirac distribution and $s$ a scaling constant. Frasca and Khurshudyan (2019) proved that if $N$ satisfies the generalized multiplicativity property $N(\theta\cdot w)=\theta(t)\cdot N(w)$, where $\theta$ is the Heaviside step function, then $G(t)=\theta(t)\,w_0(t)$ with $w_0$ the general solution of the homogeneous equation; this was established for $N(w)=w^n$ (integer $n$, extended to real $n>0$ at the a.e./distributional level by Khurshudyan 2020), $\sinh$, $\tanh$, $\sin$, $\tan$, $\arcsin$, $\arctan$, $\mathrm{arcsinh}$, $\mathrm{arctanh}$, $\ln(1+w)$ and combinations, while the Liouville case $N(w)=\exp w$ fails the property and yields a Green's function built from two one-sided contributions. Determine, within an explicitly specified theory of nonlinear distributions in which $N(G)$ is meaningful (e.g. a Colombeau-type algebra of generalized functions):

1. For the eight transcendental nonlinearities $N\in\{\cos,\cosh,\cot,\coth,\arccos,\mathrm{arccot},\mathrm{arccosh},\mathrm{arccoth}\}$ — the families the authors observed to behave like the Liouville case — determine the structure of the distributional Green's functions of $G''+N(G)=s\,\delta(t)$, together with their finite combinations in the sense of the source paper's closure convention (Remark 3 of Frasca & Khurshudyan 2019: finite combinations obtained by adding a linear term, taking linear combinations, and taking products of integer powers), at the almost-everywhere/distributional level of rigor used in the source. The generalized multiplicativity property is already settled and is not part of the question: it requires $N$ to be defined at the origin with $N(0)=0$, which none of these eight functions satisfies, so the representation $G=\theta\,w_0$ does not apply.

References

1. Marco Frasca, Asatur Zh. Khurshudyan (2019). A general representation for the Green's function of second order nonlinear differential equations. Computational and Mathematical Methods 1(4):e1038. DOI: 10.1002/cmm4.1038. https://doi.org/10.1002/cmm4.1038
2. Marco Frasca, Asatur Zh. Khurshudyan (2018). A General Representation for the Green's Function of Second Order Nonlinear Differential Equations (author version of the preceding paper). arXiv:1805.10495. https://arxiv.org/abs/1805.10495
3. Asatur Zh. Khurshudyan (2020). An identity for the Heaviside function and its application in representation of nonlinear Green's function. Computational and Applied Mathematics 39(1):32. DOI: 10.1007/s40314-019-1011-5. https://doi.org/10.1007/s40314-019-1011-5
4. Marco Frasca (2007). Green function method for nonlinear systems. Modern Physics Letters A 22(18):1293–1299. DOI: 10.1142/S0217732307023705. https://doi.org/10.1142/S0217732307023705
5. Marco Frasca (2008). Green functions and nonlinear systems: Short time expansion. International Journal of Modern Physics A 23(2):299–308. DOI: 10.1142/S0217751X08038160. https://doi.org/10.1142/S0217751X08038160
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7. Marco Frasca, Asatur Zh. Khurshudyan (2018). Green's functions for higher order nonlinear equations. International Journal of Modern Physics C 29(11):1850104. DOI: 10.1142/S0129183118501048. https://doi.org/10.1142/S0129183118501048 (preprint arXiv:1806.11313, https://arxiv.org/abs/1806.11313)
8. Marco Frasca, Asatur Khurshudyan (2018). General Representation of Nonlinear Green's Function for Second Order Differential Equations Nonlinear in the First Derivative. arXiv:1806.00274. https://arxiv.org/abs/1806.00274
9. Md Musabbir Hossain, Asatur Zh. Khurshudyan (2019). Heuristic control of nonlinear power systems: Application to the infinite bus problem. Archives of Control Sciences. DOI: 10.24425/acs.2019.129382. https://doi.org/10.24425/acs.2019.129382