Advanced Search

Query:

Fill in the form and click »Search«...

Format:

Display: entries per page entries

Zbl 0861.35094
de Bouard, Anne
(de Bouard, Anne)
Stability and instability of some nonlinear dispersive solitary waves in higher dimension. (English)
[J] Proc. R. Soc. Edinb., Sect. A 126, No.1, 89-112 (1996). ISSN 0308-2105; ISSN 1473-7124/e

This extensive paper is concerned with stability properties of radially symmetric solitary waves solutions for nonlinear evolution equations $$\partial_t u+\Delta\partial_{x_1}u+ \partial_{x_1}(f(u))=0,\quad x=(x_1,x')\in\bbfR^n=\bbfR\times\bbfR^{n-1},\tag1$$ and $$\partial_tu-\Delta\partial_tu+(f(u))_{x_1}=0,\quad x=(x_1,x_2)\in\bbfR^2.\tag2$$ Suppose $n$ is 2 or 3, $f\in C^{s+1}(\bbfR^n)$, $s>1+2^{-1}n$, $f(0)=f'(0)=0$, $f(s)=O(|s|^{p+1})$ as $|s|\to+\infty$, and $0<p<4(n-2)^{-1}$. The author considers a smooth function in (1) of the form $u(x,t)=\varphi_c(x_1-ct,x')$, $c>0$. Then, if $\varphi_c$ and $\Delta\varphi_c$ decrease to $0$ at infinity, we have $$-c\varphi_c+ \Delta\varphi_c+f(\varphi_c)=0.\tag3$$ Under the assumption above, the equation (3) possesses a positive, radially symmetric solution $\varphi_c\in H^1(\bbfR^n)$. The function $\varphi_c$ is called stable if for all $\varepsilon>0$, there is $\delta>0$ such that, if $u_0\in U_\delta$ and $u(.,t)$ is a solution of (1), with $u(.,0)=u_0$, then $u(.,t)\in U_\varepsilon$ for all $t>0$, where $U_\varepsilon$ is the set of $u\in H^1(\bbfR^n)$ such that $\inf_{\alpha\in\bbfR^n}|u-\varphi_c(.,-\alpha)|_1<\varepsilon$.\par The main result of this paper states that if the curve $c\mapsto\varphi_c$ is $C^1$ with values in $H^2(\bbfR^n)$, there exist $C>0$, $\delta_1>0$ such that $|{d\varphi_c\over dx} (x)|\le Ce^{-\delta_1|x|}$, $x\in\bbfR^n$, and the null space of the linearized operator $L_c=-\Delta+c-f'(\varphi_c)$ is spanned by $\{\partial_{x_j}\varphi_c; 1\le j\le n\}$, then $\varphi_c$ is stable if and only if $d''(c)>0$, where $d(c)= E(\varphi_c)+cQ(\varphi_c)$ and $E(\varphi_c)= 2^{-1}\int_{\bbfR^n}|\nabla \varphi_c|^2dx- \int_{\bbfR^n} F(\varphi_c)dx$ with $F'=f$, $F(0)=0$ and $Q(\varphi_c)= 2^{-1}\int_{\bbfR^n} \varphi^2_c dx$. Moreover, if we define the stability of $\varphi_c$ for equation (2) in the same way as we did above for the equation (1), then the same result as for (1) holds for the equation (2).
[D.M.Bors (Iaşi)]

MSC 2000:: *35Q51 Solitons
35B35 Stability of solutions of PDE
35Q30 Stokes and Navier-Stokes equations
76B15 Wave motions (fluid mechanics)

Keywords: radially symmetric solitary waves; nonlinear evolution equations

Comment on this Item PDF MathML XML ASCII DVI PS BibTeX Online Ordering Article Journal Journal

	Scientific prize winners of the ICM 2010
	Overhang
	Lie groups, physics and geometry. An introduction for physicists, engineers and chemists.

Advanced Search

Zentralblatt MATH Berlin [Germany]