TY - JOUR

T1 - On the modified Korteweg-De Vries equation

AU - Hayashi, Nakao

AU - Naumkin, Pavel

N1 - Funding Information:
★ Present address: Department of Mathematics, Osaka University, Osaka 560-0043, Japan. e-mail: nhayashi@math.wani.osaka-u.ac.jp ★★ The work of P.N. is partially supported by CONACYT.

PY - 2001

Y1 - 2001

N2 - We consider the large time asymptotic behavior of solutions to the Cauchy problem for the modified Korteweg-de Vries equation ut + a(t)(u3)x + 1/3uxxx = 0, (t, x) ε R × R, with initial data u(0, x) = u0(x), x ε R. We assume that the coefficient a(t) ε C1(R) is real, bounded and slowly varying function, such that |a′(t)| ≤ C〈t〉-7/6, where 〈t〉 = (1 + t2)1/2. We suppose that the initial data are real-valued and small enough, belonging to the weighted Sobolev space H1,1 = {φ ε L2; || √1+x2 √1-∂x2φ|| < ∞}. In comparison with the previous paper (Internat. Res. Notices 8 (1999), 395-418), here we exclude the condition that the integral of the initial data u0 is zero. We prove the time decay estimates 3√t2 3√〈t〉||u(t)ux(t)||∞ ≤ Cε and 〈t〉1/3-1/3β||u(t)||β ≤ Cε for all t ε R, where 4 < β ≤ ∞. We also find the asymptotics for large time of the solution in the neighborhood of the self-similar solution.

AB - We consider the large time asymptotic behavior of solutions to the Cauchy problem for the modified Korteweg-de Vries equation ut + a(t)(u3)x + 1/3uxxx = 0, (t, x) ε R × R, with initial data u(0, x) = u0(x), x ε R. We assume that the coefficient a(t) ε C1(R) is real, bounded and slowly varying function, such that |a′(t)| ≤ C〈t〉-7/6, where 〈t〉 = (1 + t2)1/2. We suppose that the initial data are real-valued and small enough, belonging to the weighted Sobolev space H1,1 = {φ ε L2; || √1+x2 √1-∂x2φ|| < ∞}. In comparison with the previous paper (Internat. Res. Notices 8 (1999), 395-418), here we exclude the condition that the integral of the initial data u0 is zero. We prove the time decay estimates 3√t2 3√〈t〉||u(t)ux(t)||∞ ≤ Cε and 〈t〉1/3-1/3β||u(t)||β ≤ Cε for all t ε R, where 4 < β ≤ ∞. We also find the asymptotics for large time of the solution in the neighborhood of the self-similar solution.

KW - Large time asymptotics

KW - Modified korteweg-de vries equation

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U2 - 10.1023/A:1012953917956

DO - 10.1023/A:1012953917956

M3 - Article

AN - SCOPUS:1542666138

SN - 1385-0172

VL - 4

SP - 197

EP - 227

JO - Mathematical Physics Analysis and Geometry

JF - Mathematical Physics Analysis and Geometry

IS - 3

ER -