Asymptotic Expansions for One-Phase Soliton-Type Solutions of the Korteweg-De Vries Equation with Variable Coefficients

Asymptotic two-phase solitonlike solutions of the singularly perturbed Korteweg-de Vries equation with variable coefficients

“…By analogy with [7,8], the singular part of asymptotics (4) is determined on each curve x = ϕ s t ( ) , s = 1, 2. The functions V t 0 1 2 ( , , ) τ τ are found by using the following homogeneous quasilinear partial differential equation: …”

“…In finding the solution of system (17), it is a priori assumed that the required solution satisfies the compatibility conditions (7). Note that the system of equations (17) possesses solutions of this sort, which follows from the presence of the two-soliton solution (2) of the Korteweg -de Vries equation (1), which is a special case of Eq.…”

“…Therefore, in numerous works, the problem of construction of solutions of this sort is solved by using asymptotic methods. Thus, in particular, in [5][6][7][8][9][10][11][12], the problem of construction of asymptotic solutions of the Korteweg -de Vries equation with variable coefficients is studied for various types of singularities. For some values of the coefficients, these solutions turn into soliton solutions and, hence, they are called solitonlike solutions [5,6].…”

Asymptotic solutions of the Cauchy problem for the singularly perturbed Korteweg-de Vries equation with variable coefficients

“…If the functions F t j ( , ) τ 1 , j N = 1, , satisfy the condition F j t G ( , ) τ 1 1 0 ∈ , j = 1, N , then, for the solvability of the operator equations (18) in the space G 1 , it is necessary and sufficient that the orthogonality condition [19] …”

“…(20) in the space G 1 0 , it is necessary and sufficient that the following orthogonality condition [21] be satisfied:…”