Complete vs General Integral of first order PDE

In Russian "integral" is a synonym of a solution of differential equation. "general integral" means general solution, "complete" probably means sum of particular solution and general solution (called the complementary solution)


In the general theory of partial differential equations and specifically for First-Order Partial Differential Equations one defines the general solution(Landau's general integral) and the complete integral as follows:

For a two-dimensional first order partial differential equation $$f(x,y,z,z_x,z_y)=0. \tag{1}$$ Complete Integral: A two parameter family of implicit solutions of the form (2) of (1) is called a complete integral of the partial differential equation. $$\phi(x,y,z,a,b)=0. \tag{2}$$ General solution: A function of the form (3), where $u(x,y,z)$ and $v(x,y,z)$ are functions of $x,y,z$ and $\Phi$ is an arbitrary smooth function, $\Phi$ is called a (implicit or explicit) general solution of (1), if $z,z_x,z_y$ as determined by the relation (3) satisfy (1) $$\Phi(u,v)=0\tag{3}.$$

*If we have a complete integral (2) of (1), we can derive a general solution (3), we would show this later in the post but first, let's see how to derive the PDE (1) from the complete integral (2).

If we have a complete integral (2), we can obtain $d\phi/dx$ and $d\phi/dy$ : $$\phi_x+z_x\phi_z=0. \tag{4}$$ $$\phi_y+z_y\phi_z=0. \tag{5}$$

With (2),(4),(5) we can obtain an expression of the form (1) that is free from the parameters $a$ and $b$. If (1) is obtained exactly from (2),(4),(5) then $\phi$ is a solution of the PDE (1).

Now, to derive a general solution (3) from a complete integral (2), we can impose $b=W(a)$ in the complete solution (2), obtaining $\Phi(x,y,z,a,W(a))$, and impose the condition $d\Phi/da=0$, $$\frac{d\Phi}{da}=\Phi_a(x,y,z,a,W(a))+W'(a)\Phi_W(x,y,z,a,W(a))=0. \tag{6}$$

With (6) we can write $a=A(x,y,z)$ as a function of $x,y,z$. So the general solution derived from (2) can be written as $$\Phi\Big(x,y,z,A(x,y,z),W(A(x,y,z))\Big) = 0. \tag{7}$$

We can see that (7) in fact matches our definition of general solution.Now we will prove that (7) is a solution of (1), again $d\Phi/dx$ and $d\Phi/dy$

$$\Phi_x+z_x\Phi_z+ \Phi_A A_x+\Phi_W W'(A) A_x =0. \tag{8}$$ $$\Phi_y+z_y\Phi_z+ \Phi_A A_y+\Phi_W W'(A) A_y =0. \tag{9}$$

Now applying the condition (6) Equations (8), and (9) yield:

$$\Phi_x+z_x\Phi_z =0. \tag{8}$$ $$\Phi_y+z_y\Phi_z =0. \tag{9}$$

Now the systems of equations (2),(4),(5) yield the same derived expression (1) as (7),(8),(9), now $\Phi\Big(x,y,z,A(x,y,z),W(A(x,y,z))\Big)$ is a general solution of (1) and we can see that we obtain a different solution for every function $W$.

We can see that this solution is free of the parameters $a$ and $b$, when we choose a particular function $W$ we obtain a particular solution for the PDE.

Landau's generalizes this result in his footnote, however he does it for an easier equation, not a general First-order PDE (1). The steps he does are the same as we did for a general two dimensional First-order PDE.

The notion "complete integral" here refers to solutions of specific (1st order) PDEs that depend on the maximal number of constants of motion. If you want a concrete example I can refer you to Equation (10) of this paper, or even better to Ref. [10] in that paper.

A "general solution", by contrast, need not depend explicitly on constants of motion, but usually contains some free (integration) function. As an example for a solution not depending explicitly on constants of motion see the "enveloping solution" in Eq. (11) of the paper above.

[I never encountered these notions anywhere, except when solving Hamilton-Jacobi equations - this seems to be also the context to which Landau refers.]