Obtaining a Lagrangian from a NLPDE The variational approximation is often used when working with optical solitons. Similarly, the Noether theorem is often used to find conserved quantities. Both of these methods require a Lagrangian. However, not every soliton-equation has a Lagrangian (hence methods like Hasegawa's).
Is there any way to check if a given non-linear partial differential equation corresponds to the Euler Lagrange equation of a Lagrangian?
Take, for example, the equation:
$(u_{t}+6uu_{x})_{x}+3 \sigma^{2}u_{yy}=0$
which is known as the dispersionless KP equation. What should I do to know if it corresponds to the Euler-Lagrange equation of a Lagrangian?
 A: Indeed, this is not always an easy task. There are some mathematical tools aimed at retrieving Lagrangian densities from (partial) differential equations such as the semi-inverse method, see Ref. (1). Note that there might be several possibilities to derive consistent variational principles, see e.g. Ref. (2).
In variational calculus, the choice of variables and coordinates plays a crucial role. One important aspect relates to the meaning of $(x,y)$ as spatial or material coordinates. Another aspect relates to the fact that the dispersionless KP equation is equivalent to the Khokhlov–Zabolotskaya equation up to a linear change of coordinates. To illustrate the first feature, let's tackle the particular case $\sigma \to 0$ of the inviscid Burgers' equation $u_t + 6uu_x = 0$. This PDE is the Euler-Lagrange equation corresponding to the variational principle (3)
$$
\delta \iint_{\Bbb R^2} v(x,t)^2 \text{d} x\, \text{d} t = 0 \, ,
$$
where $v = \dot x$ is the Eulerian velocity field ($x$ denotes Eulerian spatial coordinates).
In facts, we have $$
\delta (v^2) = 2v\, \delta v = 2v\, \delta \dot x = 2v\left((\delta x)_t + v (\delta x)_x\right) ,
$$
and the stationary-value condition becomes
$$
\iint_{\Bbb R^2} \delta (v^2)\, \text{d} x\, \text{d} t = -6\iint_{\Bbb R^2} \left(u_t + 6uu_x\right)\delta x\,  \text{d} x\, \text{d} t = 0 
$$
with $u=\frac13 v$.


*

*J.-H. He, "Variational Principle for the Generalized KdV-Burgers Equation with Fractal Derivatives for Shallow Water Waves", J. Appl. Comput. Mech. 6(4) (2020), 735-740. DOI:10.22055/JACM.2019.14813

*J.K. Hunter, Y. Zheng, "On a completely integrable nonlinear hyperbolic variational
equation", Physica D 79 (1994), 361-386. DOI:10.1016/S0167-2789(05)80015-6

*V.L. Berdichevsky, Variational Principles of Continuum Mechanics I: Fundamentals, Springer, 2009. DOI:10.1007/978-3-540-88467-5

