Nonlinear PDE from Chain of Oscillators Some years ago, I was reviewing the calculation for the dynamics of limiting case for a chain of springs with transverse oscillations and found a partial differential equation for which I haven't been able to find a reference. I'm curious about the validity of the steps leading to the PDE, the possible solutions (assuming said validity), and whether other similar equations are known.
The basic idea for obtaining the PDE is that when you consider the dynamics of a chain of transverse hookean oscillators, then one limit of the parameters gives you the standard wave equation, but a different limit gives you a nonlinear PDE.

First, I’ll review the standard wave equation derivation to set up for the alternative limit. We have a system of $N$ particles of mass $m$ arranged horizontally but constrained to move only vertically. The masses are separated by a horizontal distance $a$ and are coupled to their neighbors by a spring of spring constant $k$ and rest length $\ell_{R}$. Our goal is to consider the system for an $N\to \infty$ number of masses and to determine the associated dynamical equation.

For the $j$th mass, the magnitude of the force exerted by the $j+1$ neighbor is (according to Hooke's law)
\begin{equation}
|\vec{F}_{j+1 \text{ on } j}| = k \left(\sqrt{(y_{j+1}-y_{j})^2+a^2} - \ell_{R}\right).
\end{equation}
Since the masses can only move in the vertical direction, only the vertical component of this force is dynamically relevant. We can show that the sine of the angle between the horizontal axis and the spring connecting the $j$ and $j+1$ masses is
\begin{equation}
\sin \theta_{j} = \frac{y_{j+1}-y_{j}}{\sqrt{(y_{j+1}-y_{j})^2+a^2}}. 
\end{equation}
Therefore, the vertical component of the force exerted on $j$ from $j+1$ is
\begin{align}
 F_{j+1 \text{ on } j, \,y} &= |\vec{F}_{j+1 \text{ on } j}| \sin \theta_{j}  \\
 & = k \left(\sqrt{(y_{j+1}-y_{j})^2+a^2} - {\ell}_{R}\right)\cdot \frac{(y_{j+1}-y_{j})}{\sqrt{(y_{j+1}-y_{j})^2+a^2}} \\
  & = k(y_{j+1}-y_{j})\left[1-\frac{\ell_R}{a}\frac{1}{\sqrt{1+(y_{j+1}-y_{j})^2/a^2}} \right],
  \label{eq:non_osc}
\end{align}
Calculating the analogous quantity for $F_{j-1 \text{ on } j, \,y}$ and computing the equation of motion for $y_j$ we find
\begin{align}
m \ddot{y}_{j} & =F_{j+1 \text{ on } j, \,y}  + F_{j-1 \text{ on } j, \,y} \\
 & = k(y_{j+1}-y_{j})\left[1-\frac{\ell_R}{a}\frac{1}{\sqrt{1+(y_{j+1}-y_{j})^2/a^2}} \right]\\
 & \qquad -k(y_{j}-y_{j-1})\left[1-\frac{\ell_R}{a}\frac{1}{\sqrt{1+(y_{j}-y_{j-1})^2/a^2}} \right] \quad (1).
 \label{eq:eom}
\end{align}
Taking the limit $\ell_R \ll a$, we find the equation
\begin{align}
m \ddot{y}_{j} & = k(y_{j+1}-y_{j}) - k(y_{j}-y_{j-1}) + O(\ell_R/a), 
\end{align}
which when we take the continuum limit $y_j(t) \to y(x, t)$, $\lim_{a\to 0} ka = T$ and $\lim_{a\to 0} m/a = \mu$ leads to the expected wave equation
\begin{equation}
\mu \frac{\partial^2 y}{\partial t^2} = T \frac{\partial^2y}{\partial x^2}  \qquad (1).
\label{eq:linwv_eqn}
\end{equation}
Now for a different case.
If we take $\ell_R = a$ and $|y_{j} - y_{j-1}|\ll a$ for all $j$ in (1), then (with $1 -1/\sqrt{1+x^2} = x^2/2 + O(x^4)$) we obtain the equation of motion
\begin{align}
m \ddot{y}_{j} & = k(y_{j+1}-y_{j})\cdot \frac{1}{2} \frac{(y_{j+1}-y_{j})^2}{a^2}-  k(y_{j}-y_{j-1})\cdot\frac{1}{2} \frac{(y_{j}-y_{j-1})^2}{a^2}+ \cdots\\
 & = \frac{k}{2a^2} \left[ (y_{k+1}-y_k)^3  - (y_{k}- y_{k-1})^3 \right]+ \cdots.
 \label{eq:eom2}
\end{align}
Taking the continuum limit of this equation, then yields the PDE
\begin{equation}
\mu \frac{\partial^2 y}{\partial t^2} = \frac{T}{2} \frac{\partial}{\partial x} \left[ \left(\frac{\partial y}{\partial x}\right)^3\right] \qquad (2).
\label{eq:nonlinwv_eqn}
\end{equation}
Which is a nonlinear PDE for the oscillator system.
I have three questions about this result

*

*Derivation Validity Is the derivation of (2) valid? The places where I perceive a problem are in the two assumptions $\ell_R = a$ (i.e., the oscillator rest-length matches the horizontal distance between the oscillators) and $|y_{j+1}-y_j| \ll a$, especially given that we take $a\to 0$. The rest length condition is perhaps more justifiable since $\ell_R$ is a model parameter, but could we reasonably take $|y_{j+1}-y_j| \ll a$ if we ultimately take $a$ to $0$? In the derivation of (1), we took $a\to0$, but also have corrections of order $O(\ell_R/a)$ which itself might conflict with limit.


*Possible Solutions What are the possible solutions to (2)? It is clearly linear so guessing and checking exponential solutions won't work.


*Similar Equations Are there equations similar to (2) in the study of wave propagation in media? The only other one that seems similar is the Euler-Bernoulli Beam equation: https://en.wikipedia.org/wiki/Euler%E2%80%93Bernoulli_beam_theory#Dynamic_beam_equation
 A: with:
$$F_1=k \left( \sqrt {{y_{{10}}}^{2}+{a}^{2}}-l_{{R}} \right)$$
$$\sin(\theta_1)={\frac {y_{{10}}}{\sqrt {{y_{{10}}}^{2}+{a}^{2}}}}
$$
$$F_2=k \left( \sqrt {{y_{{20}}}^{2}+{a}^{2}}-l_{{R}} \right)$$
$$\sin(\theta_2)={\frac {y_{{20}}}{\sqrt {{y_{{20}}}^{2}+{a}^{2}}}}
$$
where
$$y_{10}=y_{j+1}-y_j\\
y_{20}=y_{j-1}-y_j$$
now  linearized the forces and the sin you obtain
Case I:
with $~l_R=\,a\,\epsilon~$ and $~\epsilon \ll 1$
$$F_i=k\,\sqrt{y_{i0}^2+a^2}\\
\sin(\theta_i)=\frac{y_{i0}}{\sqrt{y_{i0}^2+a^2}}\\
F_i\sin(\theta_i)=k\,y_{i0}$$
Case II
with $~l_R=a~,y_{i0}\ll 1~$ and the Taylor expansion $~O(y_{i0}^4)$
$$F_i=\frac k2\frac {y_{i0}^2}{a} \\
\sin(\theta_i)=\frac{y_{i0}}{a}\\
F_i\,\sin(\theta_i)=\frac k2\frac {y_{i0}^3}{a^2}$$
but the force that you obtain is only because the Taylor expansion of $~y_{i0}~$ is of order three .

Taylor series of $~F_2\,\sin(\theta_2)~$ with $~O(y_{i0}^6)~$ is:
$$\frac 12\,{\frac {k{{ \, y_{i0}}}^{3}}{{a}^{2}}}-\frac 38\,{\frac {k{{  \,y_{i0}}}^{5}
}{{a}^{4}}}$$
