The dissipation of nonlinear dispersive waves: the case of asymptotically weak nonlinearity

Asymptotics of Solutions to the Boundary-Value Problem for the Korteweg–de Vries–Burgers Equation on a Half-Line

“…Some results on the decay estimates of the solutions in different norms to the Cauchy problems for some nonlinear evolution equations with strongly dissipative operators were obtained in papers [1][2][3]. In papers [4,[7][8][9]] the large-time asymptotic behavior of solutions to the Korteweg-de Vries-Burgers-type equations, on the line was found. In paper [5] the problem u t + u 2 x − u xx + u xxx = 0 t>0 x > 0 u x 0 = u 0 x x > 0 u 0 t = 0 t > 0 was studied, and asymptotic behavior of solutions was obtained under weaker conditions on the data since the L 2 norm of the nonlinear term decays faster than the case of this paper.…”

Global behavior for Whitham type equations on a finite interval

“…While not attempting to provide a complete review of this publications, we do list some known results [1,2,[5][6][7][8][9][10][11][12][13][14][15][16][17][18][19]22,23,[27][28][29]34,[30][31][32][33][35][36][37].…”

Ott–Sudan–Ostrovskiy type equations on a segment with large initial data

“…7 648 E. Kaikina ZAMP Here and below p α is the main branch of the complex analytic function in the half-complex plane ℜp ≥ 0, so that 1 α = 1 .We make a cut along a contour Γ Γ = z ∈ C, z ∈ ∞e i(−2π+β) , 0e i(−2π+β) ∪ 0e iβ , ∞e iβ , (1.2) that is we choose arg z ∈ [−2π + β, β) for any complex z ∈ C, β ∈ π 2 , 3π 2 . Recently, much attention has been drawn to the study of the global existence and large-time asymptotic behavior of solutions to the Cauchy problem for nonlinear evolution equations (see,for example, [2]- [10], [12], [13], [15]- [18], [21], [22], [27], [28], [30]- [39]). Nonlinear pseudodifferential equations on a half-line were studied in the book [20].…”