Actually I am a little bit confused about the definition. I have read two three articles but I could not find out what type of equations are called a non-linear partial differential equation. Articles are following.
$pq = 0$ will be a first order non linear Partial differential equation? p,q are usual notation in PDE.
Please don' downvote. I know it is a silly question. But I am really confused. Please help me. I am looking forward to ur reply.
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$\begingroup$A nonlinear pde is a pde in which either the desired function(s) and/or their derivatives have either a power $\neq 1$ or is contained in some nonlinear function like $\exp, \sin$ etc, or the coordinates are nonlinear. for example, if $\rho:\mathbb{R}^4\rightarrow\mathbb{R}$ where three of the inputs are spatial coordinates, then an example of linear: $$\partial_t \rho = \nabla^2\rho$$ and now for nonlinear nonlinear $$\partial_t \rho = \nabla^2\rho+ \cos\rho$$
As I stated at the beginning A nonlinear pde can also be a pde in which the coordinates are non linear. Example:: $$\partial_t \rho(x,y,z,t)= \nabla^2 \rho(x,y,z,t)+xy-yz $$ the $xy$ and $yz$ make it nonlinear. P and q are analogous to x y z and/or t. $$ \partial_t \rho(x,y,z,t)= \nabla^2 \rho(x,y,z,t)+x^{\frac{13}{21}}$$Is also nonlinear.
In your notation, Example:: $$\partial_t \rho(p,q)= \nabla^2 \rho(p,q)+pq$$ is nonlinear due to $pq$
$\endgroup$ $\begingroup$Here's an example for finding geodesics on an arbitrary manifold:$$\ddot{x}^k=-\Gamma_{ij}^k\dot{x}^i\dot{x}^j.$$
$\endgroup$ 2 $\begingroup$What you are looking for are functions $F(x,y,z,p,q)$ that are non-linear in $p$ and $q$ and that define a first order partial differential equation via$$ 0=F(x,y,u(x,y),u_x(x,y),u_y(x,y)). $$The usual trick is to establish the Lagrange-Charpit equations $$ ds=\frac{dx}{F_p}=\frac{dy}{F_q}=\frac{dz}{pF_p+qF_q}=-\frac{dp}{F_x+pF_z}=-\frac{dq}{F_y+qF_z}. $$for the characteristic curves and assemble a family of them to form the solution surface $z=u(x,y)$.
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