The Burgers-FKPP advection-reaction- diffusion equation with cut-off

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N. Popovic, M. Ptashnyk and Z. Sattar,The Burgers-FKPP advection-reaction- diffusion equation with cut-off, J. Dyn. Differ. Equations (online first), DOI https://doi.org/10.1007/s10884-025-10458-y, 27p

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the last two equations yieldp x v =−D(u)u x andp x u =−D(u)v x. Moreover, the equation for ubecomes −∂tu−∂ xpx v =C(u)− B(u) D(u) px v . Substitutingp x v =−D(u)u x, we obtain ut −∂ x D(u)u x
+B(u)u x +C(u) = 0.(20) Hence, this two-contact Hamiltonian system induces a whole family of generalized Burgers- type equations of convection-diffusion-reaction type [55]. Several relevant cases are recovered immediately. IfD(u) =ν >0 is constant and C(u) = 0, then (20) reduces tou t +B(u)u x =νu xx. In particular, forB(u) =uone recovers the viscous Burgers equationu t +u u x =νu xx (see for instance [63]). More generally, if D(u) =νis constant, then (20) becomesu t +B(u)u x =νu xx −C(u), which includes Burgers-type equations with reaction terms, namely withC(u)̸= 0 (cf.

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the last two equations yieldp x v =−D(u)u x andp x u =−D(u)v x. Moreover, the equation for ubecomes −∂tu−∂ xpx v =C(u)− B(u) D(u) px v . Substitutingp x v =−D(u)u x, we obtain ut −∂ x D(u)u x
+B(u)u x +C(u) = 0.(20) Hence, this two-contact Hamiltonian system induces a whole family of generalized Burgers- type equations of convection-diffusion-reaction type [55]. Several relevant cases are recovered immediately. IfD(u) =ν >0 is constant and C(u) = 0, then (20) reduces tou t +B(u)u x =νu xx. In particular, forB(u) =uone recovers the viscous Burgers equationu t +u u x =νu xx (see for instance [63]). More generally, if D(u) =νis constant, then (20) becomesu t +B(u)u x =νu xx −C(u), which includes Burgers-type equations with reaction terms, namely withC(u)̸= 0 (cf