Bifurcation of nonplanar travelling waves in a free boundary problem

✍ Scribed by Claude-Michel Brauner; Alessandra Lunardi

• Publisher: Elsevier Science
• Year: 2001
• Tongue: English
• Weight: 143 KB
• Volume: 44
• Category: Article
• ISSN: 0362-546X
• DOI: 10.1016/s0362-546x(99)00262-x

✦ Synopsis

2 -X * ˜ = +∞. Note that ˜ does not cross the imaginary axis as u * crosses u c * . The purpose of this paper is to prove that in the dimension n = 2, there exists a sequence of bifurcation points u k * ↓ u c * giving rise to branches of nonplanar travelling waves (c; s; U ) bifurcating from the "trivial branch" (c 0 ; 0; U 0 ). Their speeds c depend on the corrugation of their fronts: if = (-1; 1) then

(1.5)

The sequence u k * is determined by the relation ˜ (u k * ) = k , wherek = -k 2 2 =4 is the ordered sequence of the negative eigenvalues of the second-order derivative in (-1; 1) associated with the Neumann boundary condition (see Fig. ).

The fact that a sequence of bifurcation points accumulates at u c * contributes to the understanding of the sharp instability phenomenon occurring at u * = u c * .

1 In papers [2,3] one can ÿnd a di erent study of existence and stability in the dimension n = 1.