TY - JOUR
T1 - Reduction of weakly nonlinear parabolic partial differential equations
AU - Chiba, Hayato
N1 - Funding Information:
The author would like to thank Professor Yoshihisa Morita and Shin-Ichiro Ei for useful comments. This work was supported by grant-in-aid for Young Scientists (B), Grant No. 22740069 from MEXT Japan.
PY - 2013/10/1
Y1 - 2013/10/1
N2 - It is known that the Swift-Hohenberg equation ∂u/∂t=-(∂x2+1)2+1)u+e{open}(u-u3) can be reduced to the Ginzburg-Landau equation (amplitude equation) ∂A/∂t=4∂x2A+e{open}(A-3A|A|2) by means of the singular perturbation method. This means that if e{open} > 0 is sufficiently small, a solution of the latter equation provides an approximate solution of the former one. In this paper, a reduction of a certain class of a system of nonlinear parabolic equations ∂u/∂t=Pu+e{open}f(u) is proposed. n amplitude equation of the system is defined and an error estimate of solutions is given. Further, it is proved under certain assumptions that if the amplitude equation has a stable steady state, then a given equation has a stable periodic solution. In particular, near the periodic solution, the error estimate of solutions holds uniformly in t > 0.
AB - It is known that the Swift-Hohenberg equation ∂u/∂t=-(∂x2+1)2+1)u+e{open}(u-u3) can be reduced to the Ginzburg-Landau equation (amplitude equation) ∂A/∂t=4∂x2A+e{open}(A-3A|A|2) by means of the singular perturbation method. This means that if e{open} > 0 is sufficiently small, a solution of the latter equation provides an approximate solution of the former one. In this paper, a reduction of a certain class of a system of nonlinear parabolic equations ∂u/∂t=Pu+e{open}f(u) is proposed. n amplitude equation of the system is defined and an error estimate of solutions is given. Further, it is proved under certain assumptions that if the amplitude equation has a stable steady state, then a given equation has a stable periodic solution. In particular, near the periodic solution, the error estimate of solutions holds uniformly in t > 0.
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U2 - 10.1063/1.4824014
DO - 10.1063/1.4824014
M3 - Article
AN - SCOPUS:84886850714
SN - 0022-2488
VL - 54
JO - Journal of Mathematical Physics
JF - Journal of Mathematical Physics
IS - 10
M1 - 101501
ER -