The large-time development of the solution to an initial-value problem for the Korteweg–de Vries–Burgers equation, II: Initial data has a discontinuous expansive step

2010 ◽  
Vol 72 (6) ◽  
pp. 2787-2802
Author(s):  
J.A. Leach
Keyword(s):  
Initial Data ◽  
Initial Value Problem ◽  
Large Time ◽  
Burgers Equation ◽  
Time Development ◽  
Initial Value ◽  
De Vries

Large-time behaviour of solutions of Burgers' equation

2002 ◽  
Vol 132 (4) ◽  
pp. 843-878 ◽  
Author(s):  
Daniel B. Dix
Keyword(s):  
Initial Data ◽  
Asymptotic Behaviour ◽  
Initial Value Problem ◽  
Large Time ◽  
Burgers Equation ◽  
Time Behaviour ◽  
Initial Value ◽  
Large Time Behaviour ◽  
Image Position ◽  
Sharp Error

The large-time asymptotic behaviour of real-valued solutions of the pure initial-value problem for Burgers' equation ut + uuxuxx = 0, is studied. The initial data satisfy u0(x) ~ nx as |x| , where n R. There are two constants of the motion that affect the large-time behaviour: Hopf considered the case n = 0 (i.e. u0L1(R)), and the case sufficiently small was considered by Dix. Here we completely remove that smallness condition. When n < 1, we find an explicit function U(), depending only on and n, such that uniformly in . When n 1, there are two different functions U() that simultaneously attract the quantity t12u(t12, t), and each one wins in its own range of . Thus we give an asymptotic description of the solution in different regions and compute its decay rate in L. Sharp error estimates are proved.

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