Application for Differential Equation of higher order [closed]

Question

We found some interesting insights in differential equations of the form

$y^{(n)}(x)+F_\lambda(y(x),y'(x),...,y^{(n-1)}(x))=0$,

i.e. for ordinary differential equations of $n$-th order with $n\geq2$. The function $F$ is polynomial which can include a set of parameters $\lambda$.

We know, that in physics usually the highest derivative is of order two(?), but we are searching for applications of this kind of differential equations for $n\geq3$ in physics, engineering, or in any other area. If you have an idea or know models or theories in which such equations occur, you input would be appreciated very much. Thanks for input and best wishes!

Answer 1

The stationary / travelling wave / soliton regime of the KdV equation and its cousins give a lot of examples. For the original KdV, under the travelling wave ansatz we have the third order equation $$ f''' - c f' + 6f f' = 0 .$$

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The Euler-Bernoulli beam theory has an equation of the form $$ [ \alpha f'']'' = F $$ If you somehow have a nonlinear load function $F$, or a nonlinear dependence of the flexural rigidity $\alpha$, then you will get a fourth order equation. (I make no claims on the physicality of such assumptions.)

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As an aside, as you mentioned that most physical models start out as second order. The appearance of the higher order derivatives usually comes from the approximation of the original higher dimensional physical model (in the form of a partial differential equation) by a simplified model (in lower dimensions, often now an ODE), with the higher order derivatives arising as a consequence of the constraints under which the approximations are derived. Both examples above are reductions of more primitive systems of partial differential equations: KdV comes from the equations of fluids after a reduction to one dimensions and imposing a shallow-water assumption; Euler-Bernoulli theory comes from the equations for elasticity under a one dimension reduction and certain assumptions on how the beam bends.

Answer 2

The Biharmonic nonlinear Schrodinger equation is of the form $$i\psi_t (t,x) + \Delta ^2 \psi + |\psi |^{2\sigma} \psi = 0$$ for some $\sigma >0$, with $x\in \mathbb{R}^d$ (d=1,2,3) and usually an initial condition $\psi(t=0,x) = \psi_0(x) \in H^2$.

For $d=1$, you look for solitary waves in this equation, then by the solitary wave ansatz $\psi(t,x) = e^{i\omega t}R(x)$ you get an ODE of the sort you were looking for -

$$-\omega R + R^{(4)} +|R|^{2\sigma}R = 0$$

Sometime we also look at the Biharmonic NLS (BNLS) as a pertubation for the NLS, something of the following form:

$$i\psi _t + \Delta \psi - \epsilon \Delta ^2 \psi + |\psi|^{2\sigma} \psi = 0 \, ,$$where $\epsilon \ll 1$. In this case, solitary waves can also be considered in the same manner as before.