3
$\begingroup$

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.

$\endgroup$
  • $\begingroup$ 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 May 11 '13 at 11:59
  • $\begingroup$ semi-linear is not linear. $\endgroup$ – TCL May 11 '13 at 13:30
  • $\begingroup$ @TCL Not linear and linear were never mentioned, or do you imply that ''not linear'' is equivalent to nonlinear? $\endgroup$ – Karlis Olte May 11 '13 at 13:47
  • 1
    $\begingroup$ It is nonlinear b/c of $uu_x$, not b/c of $u_{xxt}$. $\endgroup$ – TCL May 11 '13 at 14:48
  • $\begingroup$ @TLC 2nd equation also has $uu_x$, but it was claimed as semi-linear not nonlinear, why? $\endgroup$ – Karlis Olte May 11 '13 at 16:00
18
$\begingroup$

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?

$\endgroup$
  • $\begingroup$ For generalization, the second order and linear would be $a(x,y)\partial_{xx} u+b(x,y)\partial_{yy}u=c(x,y)u+f(x,y)$ ? $\endgroup$ – Isa Mar 31 '18 at 20:32

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.