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Register a new userThe good Boussinesq equation: long-time asymptotics
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We consider the initial-value problem for the ``good Boussinesq equation on the line. Using inverse scattering techniques, the solution can be expressed in terms of the solution of a $3 times 3$-matrix Riemann-Hilbert problem. We establish formulas for the long-time asymptotics of the solution by performing a Deift-Zhou steepest descent analysis of a regularized version of this Riemann-Hilbert problem.
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We develop an inverse scattering transform formalism for the good Boussinesq equation on the line. Assuming that the solution exists, we show that it can be expressed in terms of the solution of a $3 times 3$ matrix Riemann-Hilbert problem. The Riemann-Hilbert problem is formulated in terms of two reflection coefficients whose definitions involve only the initial data, and it has a form which makes it suitable for the evaluation of long-time asymptotics via Deift-Zhou steepest descent arguments.
We analyze the long-time asymptotics for the Degasperis--Procesi equation on the half-line. By applying nonlinear steepest descent techniques to an associated $3 times 3$-matrix valued Riemann--Hilbert problem, we find an explicit formula for the leading order asymptotics of the solution in the similarity region in terms of the initial and boundary values.
In this paper, we study the long-time dynamics and stability properties of the sine-Gordon equation $$f_{tt}-f_{xx}+sin f=0.$$ Firstly, we use the nonlinear steepest descent for Riemann-Hilbert problems to compute the long-time asymptotics of the solutions to the sine-Gordon equation whose initial condition belongs to some weighted Sobolev spaces. Secondly, we study the asymptotic stability of the sine-Gordon equation. It is known that the obstruction to the asymptotic stability of the sine-Gordon equation in the energy space is the existence of small breathers which is also closely related to the emergence of wobbling kinks. Combining the long-time asymptotics and a refined approximation argument, we analyze the asymptotic stability properties of the sine-Gordon equation in weighted energy spaces. Our stability analysis gives a criterion for the weight which is sharp up to the endpoint so that the asymptotic stability holds.
We consider the Cauchy problem for the defocusing Schr$ddot{text{o}}$dinger (NLS) equation with finite density initial data begin{align} &iq_t+q_{xx}-2(|q|^2-1)q=0, onumber &q(x,0)=q_0(x), quad lim_{x to pm infty}q_0(x)=pm 1. onumber end{align} Recently, for the space-time region $|x/(2t)|<1$ without stationary phase points on the jump contour, Cuccagna and Jenkins presented the asymptotic stability of the $N$-soliton solutions for the NLS equation by using the $bar{partial}$ generalization of the nonlinear steepest descent method. Their asymptotic result is the form begin{align} q(x,t)= T(infty)^{-2} q^{sol,N}(x,t) + mathcal{O}(t^{-1 }). end{align} However, for the space-time region $ |x/(2t)|>1$, there will be two stationary points appearing on the jump contour, the corresponding long-time asymptotics is still unknown. In this paper, for the region $|x/(2t)|>1, x/t=mathcal{O}(1)$, we found a different asymptotic expansion $$ q(x,t)= e^{-ialpha(infty)} left( q_{sol}(x,t;sigma_d^{(out)}) +t^{-1/2} h(x,t) right)+mathcal{O}left(t^{-3/4}right),$$ whose leading term is $N$-soliton solutions; the second $t^{-1/2}$ order term is soliton-soliton and soliton-radiation interactions; and the third term $mathcal{O}(t^{-3/4})$ is a residual error from a $overlinepartial$-equation. Additionally, the asymptotic stability property for the N-soliton solutions of the defocusing NLS equation sufficiently is obtained.
We consider the initial-value problem for the Sasa-Satsuma equation on the line with decaying initial data. Using a Riemann-Hilbert formulation and steepest descent arguments, we compute the long-time asymptotics of the solution in the sector $|x| leq M t^{1/3}$, $M$ constant. It turns out that the asymptotics can be expressed in terms of the solution of a modified Painleve II equation. Whereas the standard Painleve II equation is related to a $2 times 2$ matrix Riemann-Hilbert problem, this modified Painleve II equation is related to a $3 times 3$ matrix Riemann--Hilbert problem.
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