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Entire solutions of the KPP equation
Author(s) -
Hamel F.,
Nadirashvili N.
Publication year - 1999
Publication title -
communications on pure and applied mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 3.12
H-Index - 115
eISSN - 1097-0312
pISSN - 0010-3640
DOI - 10.1002/(sici)1097-0312(199910)52:10<1255::aid-cpa4>3.0.co;2-w
Subject(s) - traveling wave , mathematics , type (biology) , mathematical analysis , boundary (topology) , nonlinear system , wave equation , mathematical physics , pure mathematics , physics , geology , paleontology , quantum mechanics
This paper deals with the solutions defined for all time of the KPP equationu t = u xx + f ( u ), 0 < u ( x,t ) < 1, ( x,t ) ∈ ℝ 2 ,where ƒ is a KPP‐type nonlinearity defined in [0,1]: ƒ(0) = ƒ(1) = 0, ƒ′(0) > 0, ƒ′(1) < 0, ƒ > 0 in (0,1), and ƒ′( s ) ≤ ƒ′(0) in [0,1]. This equation admits infinitely many traveling‐wave‐type solutions, increasing or decreasing in x . It also admits solutions that depend only on t . In this paper, we build four other manifolds of solutions: One is 5‐dimensional, one is 4‐dimensional, and two are 3‐dimensional. Some of these new solutions are obtained by considering two traveling waves that come from both sides of the real axis and mix. Furthermore, the traveling‐wave solutions are on the boundary of these four manifolds. © 1999 John Wiley & Sons, Inc.