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Register a new userSoliton resolution for the complex short pulse equation with weighted Sobolev initial data
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We employ the $bar{partial}$-steepest descent method in order to investigate the Cauchy problem of the complex short pulse (CSP) equation with initial conditions in 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 $u(x,t)$ is derived in a fixed space-time cone $S(x_{1},x_{2},v_{1},v_{2})={(x,t)inmathbb{R}^{2}: y=y_{0}+vt, ~y_{0}in[y_{1},y_{2}], ~vin[v_{1},v_{2}]}$. Based on the resulting asymptotic behavior, we prove the solution resolution conjecture of the CSP equation which includes the soliton term confirmed by $N(I)$-soliton on discrete spectrum and the $t^{-frac{1}{2}}$ order term on continuous spectrum with residual error up to $O(t^{-1})$.
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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.
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.
In this work, we employ the $bar{partial}$ steepest descent method in order to study the Cauchy problem of the cgNLS equations with initial conditions in weighted Sobolev space $H^{1,1}(mathbb{R})={fin L^{2}(mathbb{R}): f,xfin L^{2}(mathbb{R})}$. The large time asymptotic behavior of the solution $u(x,t)$ and $v(x,t)$ are derived in a fixed space-time cone $S(x_{1},x_{2},v_{1},v_{2})={(x,t)inmathbb{R}^{2}: x=x_{0}+vt, ~x_{0}in[x_{1},x_{2}], ~vin[v_{1},v_{2}]}$. Based on the resulting asymptotic behavior, we prove the solution resolution conjecture of the cgNLS equations which contains the soliton term confirmed by $|mathcal{Z}(mathcal{I})|$-soliton on discrete spectrum and the $t^{-frac{1}{2}}$ order term on continuous spectrum with residual error up to $O(t^{-frac{3}{4}})$.
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 $O(t^{-frac{3}{4}})$.
In the present paper, we study the defocusing complex short pulse (CSP) equations both geometrically and algebraically. From the geometric point of view, we establish a link of the complex coupled dispersionless (CCD) system with the motion of space curves in Minkowski space $mathbf{R}^{2,1}$, then with the defocusing CSP equation via a hodograph (reciprocal) transformation, the Lax pair is constructed naturally for the defocusing CSP equation. We also show that the CCD system of both the focusing and defocusing types can be derived from the fundamental forms of surfaces such that their curve flows are formulated. In the second part of the paper, we derive the the defocusing CSP equation from the single-component extended KP hierarchy by the reduction method. As a by-product, the $N$-dark soliton solution for the defocusing CSP equation in the form of determinants for these equations is provided.
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