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We consider the compressive wave for the modified Korteweg–de Vries equation with background constants $$c>0$$ c > 0 for $$x\rightarrow -\infty $$ x → - ∞ and 0 for $$x\rightarrow +\infty $$ x → + ∞ . We study the asymptotics of solutions in the transition zone $$4c^2t-\varepsilon t<x<4c^2t-\beta t^{\sigma }\ln t$$ 4 c 2 t - ε t < x < 4 c 2 t - β t σ ln t for $$\varepsilon >0,$$ ε > 0 , $$\sigma \in (0,1),$$ σ ∈ ( 0 , 1 ) , $$\beta >0.$$ β > 0 . In this region we have a bulk of nonvanishing oscillations, the number of which grows as $$\frac{\varepsilon t}{\ln t}.$$ ε t ln t . Also we show how to obtain Khruslov–Kotlyarov’s asymptotics in the domain $$4c^2t-\rho \ln t<x<4c^2t$$ 4 c 2 t - ρ ln t < x < 4 c 2 t with the help of parametrices constructed out of Laguerre polynomials in the corresponding Riemann–Hilbert problem.
Analysis and Mathematical Physics – Springer Journals
Published: Dec 12, 2018
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