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The mKdV equation on the half-line. (English) Zbl 1057.35050
The paper is dealing with the problem of integrability of the real modified Korteweg-de Vries (mKdV) equation defined on the half-axis, \(0\leq x <\infty\). The equation is \[ q_t - q_{xxx} +6q^2q_x = 0, \] and the boundary conditions (b.c.) at the edge point \(x=0\) are \(q=q_0(t),q_{x} =q_1(t),q_{xx}=q_2(t)\). The equation is to be solved with an initial condition, \(q(x,0)=q_0(x)\). As is well known, the initial value problem for the mKdV equation on the whole axis (\(-\infty<x<+\infty\)) is exactly solvable (in principle) by means of the inverse scattering transform. In this work, a possibility is considered to apply this method to the integration of the mixed initial boundary value problem on the half-axis. To this end, simultaneous spectral analysis is developed for two eigenvalue equations from the Lax pair providing the integrability of the mKdV equation on the whole axis. The analysis expresses the solutions in terms of solutions to a corresponding matrix Riemann-Hilbert problem in the complex plane of the spectral parameter. As a result, it is found that the mixed initial boundary value problem is integrable if the set of the b.c. is subject to a special constraint, which expresses \(q_2(t)\) in terms of \(q_0(t)\) and \(q_1(t)\). Additionally, the asymptotic form of the solution at \(t\to\infty\) is analyzed, also by means of the Riemann-Hilbert-problem technique.

MSC:
35Q53 KdV equations (Korteweg-de Vries equations)
37K15 Inverse spectral and scattering methods for infinite-dimensional Hamiltonian and Lagrangian systems
35Q15 Riemann-Hilbert problems in context of PDEs
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