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A. Fokas, A. Its (1996)
The linearization of the initial-boundary value problem of the nonlinear Schro¨dinger equationSiam Journal on Mathematical Analysis, 27
Such an IBVP can be analysed, using the unique solvability of the associated Riemann-Hilbert (R-H) problem [3]. The R-H approach is used in the work
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An initial-boundary value problem for the sine-Gordon equation in laboratory coordinatesTeor. Mat. Fiz., 92
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Исмагил Хабибуллин, I. Habibullin (1999)
Уравнение КдФ на полуоси с нулевым краевым условием@@@KdV equation on a half-line with the zero boundary condition, 119
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Initial boundary value problem for the KdV equation on a semiaxis with homogeneous boundary conditionsTeor. Mat. Fiz., 130
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A. Fokas, A. Its (1992)
An initial-boundary value problem for the sine-Gordon equation in laboratory coordinatesTheoretical and Mathematical Physics, 92
We consider the initial-boundary value problem (IBVP) for the Korteweg–de Vries equation with zero boundary conditions at x=0 and arbitrary smooth decreasing initial data. We prove that the solution of this IBVP can be found by solving two linear inverse scattering problems (SPs) on two different spectral planes. The first SP is associated with the KdV equation. The second SP is self-conjugate and its scattering function is found in terms of entries of the scattering matrix s(k) for the first SP. Knowing the scattering function, we solve the second inverse SP for finding the potential self-conjugate matrix. Consequently, the unknown object entering coefficients in the system of evolution equations for s(k,t) is found. Then, the time-dependent scattering matrix s(k,t) is expressed in terms of s(k)=s(k,0) and of solutions of the self-conjugate SP. Knowing s(k,t), we find the solution of the IBVP in terms of the solution of the Gelfand–Levitan–Marchenko equation in the first inverse SP.
Acta Applicandae Mathematicae – Springer Journals
Published: Jun 6, 2013
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