Long-Time Asymptotic Behavior for the Discrete Defocusing mKdV Equation

Abstract

In this article, we apply Deift-Zhou nonlinear steepest descent method to analyze the long-time asymptotic behavior of the solution for the discrete defocusing mKdV equatioṅwith decay initial value qn(t = 0) = qn(0),where n = 0, ±1, ±2, · · · is a discrete variable and t is continuous time variable. This equation was proposed by Ablowitz and Ladik.

Asymptotics for small data solutions of the Ablowitz-Ladik equation

“…In particular, |φ(P 2 )| ∼ ξ − π 2 3 , |φ(P 4 )| ∼ ξ + π 2 3 , and φ(P 3 ) is nonvanishing over the entire frequency domain. Thus, provided ξ is at a distance at least t −1/3 away from the degenerate frequencies, we expect that the oscillations in these terms will lead to addition cancellation, preventing them from affecting the asymptotics.…”

“…Here I (2) and I (3) contain the contribution from near the stationary points P 2 and P 3 (respectively), I (1,4) contains the contribution from the region near P 1 and P 4 , and R the remainder. We claim that I (1,4) which, taken together, give the estimate (126).…”

“…In principle, the integrability of (1) means the inverse scattering transform provides an explicit solution formula [2]. Indeed, certain modified scattering results have been found, see [3,24,[29][30][31]. Historically, the complexity of the calculations required means that the modified scattering can often only be shown if the initial data is very strongly localized.…”

Rational solutions for three semi-discrete modified Korteweg–de Vries-type equations

“…Recently, soliton solutions and Jordan-block solutions for the equation (1.2) [21] was derived through the generalized Cauchy matrix approach [22]. For the sd-mKdV equation (1.3), many approaches, such as inverse scattering transform [23,24], Darboux transformation [25,26], bilinear approach [27], discrete Jacobi sub-equation method [28], algebro-geometric approach [29], Riemann-Hilbert approach [30] and Deift-Zhou nonlinear steepest descent method [31], have been developed to construct its exact solutions.…”

$L^2$ Sobolev space bijectivity of the scattering-inverse scattering transforms related to defocusing Ablowitz-Ladik systems

“…Since then, people have applied Deift-Zhou method to long-time asymptotic analysis of many integrable systems [29][30][31][32][33][34]. Researchers also generalized Deift-Zhou method to apply it on discrete integrable systems with Schwartz initial potentials, such as Toda lattice [35,36], discrete nonlinear Schrödinger equations [37][38][39] and defocusing discrete mKdV equations [40].…”