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Register a new userAsymptotics for the Sasa--Satsuma equation in terms of a modified Painleve II transcendent
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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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The Sasa-Satsuma equation with $3 times 3 $ Lax representation is one of the integrable extensions of the nonlinear Schr{o}dinger equation. In this paper, we consider the Cauchy problem of the Sasa-Satsuma equation with generic decaying initial data. Based on the Rieamnn-Hilbert problem characterization for the Cauchy problem and the $overline{partial}$-nonlinear steepest descent method, we find qualitatively different long time asymptotic forms for the Sasa-Satsuma equation in three solitonic space-time regions: (1) For the region $x<0, |x/t|=mathcal{O}(1)$, the long time asymptotic is given by $$q(x,t)=u_{sol}(x,t| sigma_{d}(mathcal{I})) + t^{-1/2} h + mathcal{O} (t^{-3/4}). $$ in which the leading term is $N(I)$ solitons, the second term the second $t^{-1/2}$ order term is soliton-radiation interactions and the third term is a residual error from a $overlinepartial$ equation. (2) For the region $ x>0, |x/t|=mathcal{O}(1)$, the long time asymptotic is given by $$ u(x,t)= u_{sol}(x,t| sigma_{d}(mathcal{I})) + mathcal{O}(t^{-1}).$$ in which the leading term is $N(I)$ solitons, the second term is a residual error from a $overlinepartial$ equation. (3) For the region $ |x/t^{1/3}|=mathcal{O}(1)$, the Painleve asymptotic is found by $$ u(x,t)= frac{1}{t^{1/3}} u_{P} left(frac{x}{t^{1/3}} right) + mathcal{O} left(t^{2/(3p)-1/2} right), qquad 4<p < infty.$$ in which the leading term is a solution to a modified Painleve $mathrm{II}$ equation, the second term is a residual error from a $overlinepartial$ equation.
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.
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.
We argue the integrability of the generalized KdV(GKdV) equation using the Painleve test. For $d( le 2)$ dimensional space, GKdV equation passes the Painleve test but does not for $d geq 3$ dimensional space. We also apply the Ablowitz-Ramani-Segurs conjecture to the GKdV equation in order to complement the Painleve test.
We study the Cauchy problem for the Korteweg-de Vries (KdV) hierarchy in the small dispersion limit where $eto 0$. For negative analytic initial data with a single negative hump, we prove that for small times, the solution is approximated by the solution to the hyperbolic transport equation which corresponds to $e=0$. Near the time of gradient catastrophe for the transport equation, we show that the solution to the KdV hierarchy is approximated by a particular Painleve transcendent. This supports Dubrovins universality conjecture concerning the critical behavior of Hamiltonian perturbations of hyperbolic equations. We use the Riemann-Hilbert approach to prove our results.
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