Green's function for second order quasi-linear equations with homogeneous nonlinearity

Problem's Description

It has been numerically established by Marco Frasca a decade ago [3] that the second order quasi-linear non-homogeneous equations $$ \frac{d^2 w}{d t^2} + N(w) = f(t), ~~ t > 0, $$ subject to homogeneous Cauchy conditions, admits the following solution called short time expansion: $$ w(t) = \int_0^t G(t - \tau) f(\tau) d \tau + \sum_{n = 1}^\infty a_n \int_0^t G(t - \tau) f(\tau) (t - \tau)^n d \tau. $$ Here $a_n$ are determined in terms of the quantities $w^{(k)}(0)$, $G$ is the general solution of the following quasi-linear equation: $$ \frac{d^2 G}{d t^2} + N(G) = s \delta(t). $$ Here $s$ is an error correcting factor introduced from numerical considerations, $\delta$ is the Dirac function. Due to the similarity with the linear case, $G$ is referred to as nonlinear Green's function.

Further, in [2] it has been shown that the first order approximation $$ w(t) \approx \int_0^t G(t - \tau) f(\tau) d \tau $$ provides a good numerical fit to the exact solution for small $t$. Higher order terms contribute to the error correction.

In [1] it has been proved that as soon as the nonlinear term satisfies the generalized homogeneity condition $$ N(\theta \cdot w) = \theta(t) \cdot N(w), $$ where $\theta$ is the Heaviside function, then the nonlinear Green's function admits the following representation: $$ G(t) = \theta(t) w_0(t), $$ where $w_0$ is the general solution of the following Cauchy problem: $$ \frac{d^2 w_0}{d t^2} + N(w_0) = 0, ~~ t > 0, $$ $$ w_0(0) = 0, ~~ \frac{d w_0}{d t}\bigg|_{t = 0} = s. $$

Problem 1. What is the general representation of $N$ satisfying the above property?

Several particular forms of $N$ are given in [1].

Evidently, $N(w) = \exp w$ (Liouville equation) does not satisfy the above property. However, its Green's function admits the following representation: $$ G(t) = 2 \theta(t) \log\left[ \frac{\sqrt{2\varepsilon}}{\cosh(\sqrt{\varepsilon} t + \phi)} \right] + 2 \theta(-t) \log\left[ \frac{\sqrt{2\varepsilon}}{\cosh(\sqrt{\varepsilon} t - \phi)} \right], ~~ \tanh\phi = - \frac{1}{4\sqrt{\varepsilon}}. $$

Problem 2. How to determine Green's function for $N(w) = \cos w$, $\cosh$, etc.?

1. ArticleIs an originA General Representation for the Green's Function of Second Order Nonlinear Differential Equations

   • Marco Frasca,
   • Asatur Khurshudyan

   arXiv

2. Article Green function method for nonlinear systems

   • Marco Frasca

   Modern Physics Letters A (MPLA) 22, 1293–1299, 2007arXiv

3. Article Green functions and nonlinear systems: Short time expansion

   • Marco Frasca

   International Journal of Modern Physics A (IJMPA) 23, 299–308, 2008arXiv