The Camassa–Holm equation on the half-line with linearizable boundary condition

2010 ◽  
Vol 348 (13-14) ◽  
pp. 775-780 ◽  
Author(s):  
Anne Boutet de Monvel ◽  
Dmitry Shepelsky
Keyword(s):  
Boundary Condition ◽  
Half Line ◽  
Holm Equation

scholarly journals Entire solutions of the Fisher–KPP equation on the half line

2019 ◽  
Vol 31 (3) ◽  
pp. 407-422 ◽  
Author(s):  
BENDONG LOU ◽  
JUNFAN LU ◽  
YOSHIHISA MORITA
Keyword(s):  
Boundary Condition ◽  
Dirichlet Boundary Condition ◽  
Traveling Wave ◽  
Wave Solution ◽  
Dirichlet Boundary ◽  
Entire Solution ◽  
Traveling Wave Solution ◽  
Half Line ◽  
Entire Solutions ◽  
Kpp Equation

In this paper, we study the entire solutions of the Fisher–KPP (Kolmogorov–Petrovsky–Piskunov) equation ut = uxx + f(u) on the half line [0, ∞) with Dirichlet boundary condition at x = 0. (1) For any $c \ge 2\sqrt {f'(0)} $, we show the existence of an entire solution ${{\cal U}^c}(x,t)$ which connects the traveling wave solution φc(x + ct) at t = −∞ and the unique positive stationary solution V(x) at t = +∞; (2) We also construct an entire solution ${{\cal U}}(x,t)$ which connects the solution of ηt = f(η) at t = −∞ and V(x) at t = +∞.

A REACTION–DIFFUSION–ADVECTION EQUATION WITH COMBUSTION NONLINEARITY ON THE HALF-LINE

2018 ◽  
Vol 98 (2) ◽  
pp. 277-285
Author(s):  
FANG LI ◽  
QI LI ◽  
YUFEI LIU
Keyword(s):  
Boundary Condition ◽  
Wave Speed ◽  
Reaction Diffusion ◽  
Robin Boundary Condition ◽  
Travelling Wave ◽  
Half Line ◽  
Advection Equation ◽  
The Right ◽  
Combustion Nonlinearity ◽  
Robin Boundary

We study the dynamics of a reaction–diffusion–advection equation $u_{t}=u_{xx}-au_{x}+f(u)$ on the right half-line with Robin boundary condition $u_{x}=au$ at $x=0$, where $f(u)$ is a combustion nonlinearity. We show that, when $0<a<c$ (where $c$ is the travelling wave speed of $u_{t}=u_{xx}+f(u)$), $u$ converges in the $L_{loc}^{\infty }([0,\infty ))$ topology either to $0$ or to a positive steady state; when $a\geq c$, a solution $u$ starting from a small initial datum tends to $0$ in the $L^{\infty }([0,\infty ))$ topology, but this is not true for a solution starting from a large initial datum; when $a>c$, such a solution converges to $0$ in $L_{loc}^{\infty }([0,\infty ))$ but not in $L^{\infty }([0,\infty ))$ topology.

Asymptotics of solutions to a convection—diffusion equation on the half-line

2000 ◽  
Vol 130 (4) ◽  
pp. 837-853 ◽  
Author(s):  
G. Karch
Keyword(s):  
Boundary Condition ◽  
Asymptotic Expansion ◽  
Neumann Boundary Condition ◽  
Neumann Boundary ◽  
Half Line ◽  
Asymptotics Of Solutions ◽  
Convection Diffusion ◽  
Convection Diffusion Equation ◽  
Homogeneous Neumann Boundary Condition ◽  
Higher Order Terms

We study the behaviour, as t → ∞, of solutions to the convectiondiffusion equation on the half-line with the homogeneous Neumann boundary condition and with bounded initial data. The higher-order terms of the asymptotic expansion in Lp (R+) of solutions are derived.

scholarly journals Classical Integrability of the O(N) Nonlinear Sigma Model on a Half-Line

1997 ◽  
Vol 12 (16) ◽  
pp. 2825-2834 ◽  
Author(s):  
E. Corrigan ◽  
Z.-M. Sheng
Keyword(s):  
Boundary Condition ◽  
Free Boundary ◽  
Boundary Conditions ◽  
Sigma Model ◽  
Free Boundary Condition ◽  
Half Line ◽  
Nonlinear Sigma Model ◽  
Conserved Charges

The classical integrability of the O (N) nonlinear sigma model on a half-line is examined, and the existence of an infinity of conserved charges in involution is established for the free boundary condition. For the case N = 3 other possible boundary conditions are considered briefly.

The Camassa–Holm Equation on the Half-Line: a Riemann–Hilbert Approach

2008 ◽  
Vol 18 (2) ◽  
pp. 285-323 ◽  
Author(s):  
Anne Boutet de Monvel ◽  
Dmitry Shepelsky
Keyword(s):  
Half Line ◽  
Holm Equation
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