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Register a new userSoliton resolution for the Hirota equation with weighted Sobolev initial data
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In this work, the $overline{partial}$ steepest descent method is employed to investigate the soliton resolution for the Hirota equation with the initial value belong to weighted Sobolev space $H^{1,1}(mathbb{R})={fin L^{2}(mathbb{R}): f,xfin L^{2}(mathbb{R})}$. The long-time asymptotic behavior of the solution $q(x,t)$ is derived in any fixed space-time cone $C(x_{1},x_{2},v_{1},v_{2})=left{(x,t)in mathbb{R}timesmathbb{R}: x=x_{0}+vt ~text{with}~ x_{0}in[x_{1},x_{2}]right}$. We show that solution resolution conjecture of the Hirota equation is characterized by the leading order term $mathcal {O}(t^{-1/2})$ in the continuous spectrum, $mathcal {N}(mathcal {I})$ soliton solutions in the discrete spectrum and error order $mathcal {O}(t^{-3/4})$ from the $overline{partial}$ equation.
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The soliton resolution for the Harry Dym equation is established for initial conditions in weighted Sobolev space $H^{1,1}(mathbb{R})$. Combining the nonlinear steepest descent method and $bar{partial}$-derivatives condition, we obtain that when $frac{y}{t}<-epsilon(epsilon>0)$ the long time asymptotic expansion of the solution $q(x,t)$ in any fixed cone begin{equation} Cleft(y_{1}, y_{2}, v_{1}, v_{2}right)=left{(y, t) in R^{2} mid y=y_{0}+v t, y_{0} inleft[y_{1}, y_{2}right], v inleft[v_{1}, v_{2}right]right} end{equation} up to an residual error of order $mathcal{O}(t^{-1})$. The expansion shows the long time asymptotic behavior can be described as an $N(I)$-soliton on discrete spectrum whose parameters are modulated by a sum of localized soliton-soliton interactions as one moves through the cone and the second term coming from soliton-radiation interactionson on continuous spectrum.
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In this work, we employ the $bar{partial}$-steepest descent method to investigate the Cauchy problem of the Wadati-Konno-Ichikawa (WKI) equation with initial conditions in weighted Sobolev space $mathcal{H}(mathbb{R})$. The long time asymptotic behavior of the solution $q(x,t)$ is derived in a fixed space-time cone $S(y_{1},y_{2},v_{1},v_{2})={(y,t)inmathbb{R}^{2}: y=y_{0}+vt, ~y_{0}in[y_{1},y_{2}], ~vin[v_{1},v_{2}]}$. Based on the resulting asymptotic behavior, we prove the soliton resolution conjecture of the WKI equation which includes the soliton term confirmed by $N(mathcal{I})$-soliton on discrete spectrum and the $t^{-frac{1}{2}}$ order term on continuous spectrum with residual error up to $O(t^{-frac{3}{4}})$.
The authors compute the long-time asymptotics for solutions of the NLS equation just under the assumption that the initial data lies in a weighted Sobolev space. In earlier work (see e.g. [DZ1],[DIZ]) high orders of decay and smoothness are required for the initial data. The method here is a further development of the steepest descent method of [DZ1], and replaces certain absolute type estimates in [DZ1] with cancellation from oscillations.
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