Higher-Order Stationary Dispersive Equations on Bounded Intervals

Abstract

A boundary value problem for a stationary nonlinear dispersive equation of 2 + 1 order on an interval (0, ) was considered. The existence, uniqueness, and continuous dependence of a regular solution have been established.

Formulation of Problems for Stationary Dispersive Equations of Higher Orders on Bounded Intervals with General Boundary Conditions

“…All results established in this paper are proven for the case l = 1, see [21]. From here on, we will consider l ≥ 2.…”

“…Therefore, precise mathematical analysis of mixed problems in bounded domains for dispersive equations is welcome and attracts attention of specialists in this area [2,[11][12][13][15][16][17][18] . Last years, publications on stationary and evolution dispersive equations of higher orders appeared [1,15,[19][20][21][22] . Usually, simple boundary conditions at x = 0 and x = L such as D i u(0) = D i u(L) = D l u(L) = 0, i = 0,..., l − 1 for (1) were imposed, see [1,21,22].…”

“…Last years, publications on stationary and evolution dispersive equations of higher orders appeared [1,15,[19][20][21][22] . Usually, simple boundary conditions at x = 0 and x = L such as D i u(0) = D i u(L) = D l u(L) = 0, i = 0,..., l − 1 for (1) were imposed, see [1,21,22]. Different kind of boundary conditions for KdV and Kawahara equations was considered in [16,[23][24][25].…”

“…Moreover, these results can be used for constructing of numerical schemes for studying various models of initialboundary value problems for higher-order dispersive equations depending on a choise of l and coefficients. Some techniques proposed in this manuscript are used to solve problems involving dispersive equations [21,22,24,25] moreover, they can be found in different problems of mathematical physics [27][28][29] .…”

“…with a constant C depending only on L, l, λ, a l+ j,l− j , b l+j,l−j , (see details in [21], p.4-5). Uniqueness of u follows from (35).…”