Andreiev K. The rarefaction waves for the Korteweg–de Vries equation: asymptotics and the integrals of motion

The thesis is concerned with the Cauchy problem solution for the Korteweg–de Vries equation with steplike initial data associated with the rarefaction waves. The long time asymptotic behavior of this solution is discussed in the regime when the ratio of the spatial and the time variables changes slowly. By use of the nonlinear steepest descent method, the asymptotics of the solution was investigated in all principal regions of the space-time half-plane: in the soliton region, in the middle region between the leading and rear wave fronts, and in the region behind the rear wave front. The result generalizes previously known results. The method is applied to studying of the vector Riemann-Hilbert (RH) oscillation problems associated with the right and left scattering data of the initial profile. The unique solvability of these RH problems is proved, in particular, in the presence of the discrete spectrum and possible resonances. Using the so-called lens and g-function mechanisms, as far as standard conjugation and deformation methods, these RH problems amount to the equivalent RH problems, with the jump matrices slightly differing from constant matrices when the time is large, except of small vicinities of finite number of the extrema points. The RH problems with the constant jump matrices (model problems) are solved explicitly. The additional parametrix RH problems are also solved, and the concluding asymptotic analysis is given. It justifies rigorously the asymptotic expansions for the rarefaction wave of the KdV equation. The regularized integrals of motion are constructed for asymptotically finite gap steplike solution for the KdV equation. In the case of the asymptotically constant initial data, the representation of the integrals of motion is given via the scattering data of the associated Schrodinger operator.