اشترك بالحزمة الذهبية واحصل على وصول غير محدود شمرا أكاديميا
تسجيل مستخدم جديدInverse scattering transform and multiple high-order pole solutions for the nonlocal focusing and defocusing modified Korteweg-de Vries equation with the nonzero boundary conditions
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We extend the Riemann-Hilbert (RH) method to study the inverse scattering transformation and high-order pole solutions of the focusing and defocusing nonlocal (reverse-space-time) modified Korteweg-de Vries (mKdV) equations with nonzero boundary conditions (NZBCs) at infinity and successfully find its multiple soliton solutions with one high-order pole and multiple high-order poles. By introducing the generalized residue formula, we overcome the difficulty caused by calculating the residue conditions corresponding to the higher-order poles. In accordance with the Laurent series of reflection coefficient and oscillation term, the determinant formula of the high-order pole solution with NZBCs is established. Finally, combined with specific parameters, the dynamic propagation behaviors of the high-order pole solutions are further analyzed and some very interesting phenomena are obtained, including kink solution, anti kink solution, rational solution and breathing-soliton solution.
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In this work, we extend the Riemann-Hilbert (RH) method in order to study the coupled modified Korteweg-de Vries equation (cmKdV) under nonzero boundary conditions (NZBCs), and successfully find its solutions with their various dynamic propagation be
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The theory of inverse scattering is developed to study the initial-value problem for the modified matrix Korteweg-de Vries (mmKdV) equation with the $2mtimes2m$ $(mgeq 1)$ Lax pairs under the nonzero boundary conditions at infinity. In the direct pro
blem, by introducing a suitable uniform transformation we establish the proper complex $z$-plane in order to discuss the Jost eigenfunctions, scattering matrix and their analyticity and symmetry of the equation. Moreover the asymptotic behavior of the Jost functions and scattering matrix needed in the inverse problem are analyzed via Wentzel-Kramers-Brillouin expansion. In the inverse problem, the generalized Riemann-Hilbert problem of the mmKdV equation is first established by using the analyticity of the modified eigenfunctions and scattering coefficients. The reconstruction formula of potential function with reflection-less case is derived by solving this Riemann-Hilbert problem and using the scattering data. In addition the dynamic behavior of the solutions for the focusing mmKdV equation including one- and two- soliton solutions are presented in detail under the the condition that the potential is scalar and the $2times2$ symmetric matrix. Finally, we provide some detailed proofs and weak version of trace formulas to show that the asymptotic phase of the potential and the scattering data.
The $n$-fold Darboux transformation $T_{n}$ of the focusing real mo-di-fied Kor-te-weg-de Vries (mKdV) equation is expressed in terms of the determinant representation. Using this representation, the $n$-soliton solutions of the mKdV equation are als
o expressed by determinants whose elements consist of the eigenvalues $lambda_{j}$ and the corresponding eigenfunctions of the associated Lax equation. The nonsingular $n$-positon solutions of the focusing mKdV equation are obtained in the special limit $lambda_{j}rightarrowlambda_{1}$, from the corresponding $n$-soliton solutions and by using the associated higher-order Taylor expansion. Furthermore, the decomposition method of the $n$-positon solution into $n$ single-soliton solutions, the trajectories, and the corresponding phase shifts of the multi-positons are also investigated.
In this paper, we consider the real modified Korteweg-de Vries (mKdV) equation and construct a special kind of breather solution, which can be obtained by taking the limit $lambda_{j}$ $rightarrow$ $lambda_{1}$ of the Lax pair eigenvalues used in the
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