Benjamin-Bona-Mahony equation: $$\displaystyle u_t+u_x+uu_x-u_{xxt}=0$$ both the paper I was reading and wikipedia claimed that it is nonlinear. It has been some time since I studied classification so I wasn't sure it is nonlinear because of $uu_x$ or $u_{xxt}$ or both.

So I looked it up and found that "... semilinear equations are ones in which the coefficients of the terms involving the highest-order derivatives of u depend only on x, not on u or its derivatives." And claims that $$\displaystyle u_t+u_{xxx}+u u_{x}=0$$ is semiliner.

So it seems to me that if BBM is indeed nonlinear, it is not because of $uu_{x}$, but because of $u_{xxt}$, but that feels unnatural for me.

Please shed some light on me.

  • $\begingroup$ the quoted definition seems incomplete. Some coefficient depends on $u$ so that makes it nonlinear, if the coefficient is for a lower order derivative then the classification relaxes to semi-linear. $\endgroup$
    – Maesumi
    Commented May 11, 2013 at 11:59
  • $\begingroup$ semi-linear is not linear. $\endgroup$
    – TCL
    Commented May 11, 2013 at 13:30
  • $\begingroup$ @TCL Not linear and linear were never mentioned, or do you imply that ''not linear'' is equivalent to nonlinear? $\endgroup$ Commented May 11, 2013 at 13:47
  • 1
    $\begingroup$ It is nonlinear b/c of $uu_x$, not b/c of $u_{xxt}$. $\endgroup$
    – TCL
    Commented May 11, 2013 at 14:48
  • $\begingroup$ @TLC 2nd equation also has $uu_x$, but it was claimed as semi-linear not nonlinear, why? $\endgroup$ Commented May 11, 2013 at 16:00

1 Answer 1


Consider first order PDE depending on two independent variables. It is

  • linear, if it has the form $$ a(x,y)\partial_x u+b(x,y)\partial_yu=c(x,y)u+f(x,y), $$ Example: $$ \partial_xu+\partial_yu=0. $$
  • semi-linear if it has the form $$ a(x,y)\partial_x u+b(x,y)\partial_yu=c(x,y,u), $$ Example: $$ \partial_xu+\partial_yu=u^2 $$
  • quasi-linear if it has the form $$ a(x,y,u)\partial_x u+b(x,y,u)\partial_yu=c(x,y,u), $$ Example: $$ \partial_xu+u\partial_yu=0 $$
  • fully non-linear if it has the form $$ F(x,y,u,\partial_xu,\partial_yu)=0. $$ Example: $$ (\partial_xu)^2+(\partial_yu)^2=V(x,y). $$

Note that semi-linear, quasi-linear, and fully nonlinear equations are nonlinear.

Can you generalize this classification to equations of any order?

  • $\begingroup$ For generalization, the second order and linear would be $a(x,y)\partial_{xx} u+b(x,y)\partial_{yy}u=c(x,y)u+f(x,y)$ ? $\endgroup$
    – user486983
    Commented Mar 31, 2018 at 20:32

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