New Exact Superposition Solutions to KdV2 Equation

Abstract

New exact solutions to the KdV2 equation (also known as the extended KdV equation) are constructed. The KdV2 equation is a second-order approximation of the set of Boussinesq's equations for shallow water waves which in first-order approximation yields KdV. The exact solutions ( /2)(dn) + in the form of periodic functions found in the paper complement other forms of exact solutions to KdV2 obtained earlier, that is, the solitonic ones and periodic ones given by single cn 2 or dn 2 Jacobi elliptic functions.

Martingale Solution to Stochastic Extended Korteweg-de Vries Equation

“…what reflects finitness of solutions to deterministic version of the equation (2.1) (see, e.g., [10,21,22]). The operator Φ is a continuous mapping from H 2 (R) to L 0 2 (L 2 (R)), the space of Hilbert-Schmidt operators from L 2 (R) to itself.…”

Inverted solutions of KdV-type and Gardner equations

“…The equation ( 18), unlike the KdV equation ( 1), is nonintegrable. Despite this fact, it has analytic single-soliton solutions [8], as well as periodic cnoidal solutions [1] and superposition solutions [3,4]. However, multi-soliton solution to (18) do not exist [9].…”

“…The analytic single-soliton and periodic solutions of the KdV2 equation have the same functional form as the analogous solutions of the KdV equation but the corresponding coefficients B, v are slightly different, and the coefficient A is determined by the parameters of the equation, i.e., by α, β. It is because the KdV2 equation imposes one more condition on the coefficients A, B, v, determining the solution, than the KdV equation [1,3,4,5,8]. For KdV, the set of these coefficients has one degree of freedom, which, for fixed α, β, allows solitons of different heights to exist and hence admits multi-soliton solutions.…”

Remarks on Existence/Nonexistence of Analytic Solutions to Higher Order KdV Equations

“…In this section, we discuss the solutions to KdV2 equation (2) derived by us in the same way as solutions to KdV (1) in section 2. In [9][10][11][12][13], we have shown that for the KdV2 equation there exist analytic solutions of the same forms as solutions to KdV (4), (8) and (13) but with different coefficients A, B, D, v. Here, we give a brief overview of these results, the full presentation of which is contained in [9][10][11][12][13].…”

“…Finally, the 'superposition' solutions to KdV2 are qualitatively similar to cnoidal solutions, but with slightly different amplitudes, velocities and wavelengths. For more details, we refer to [11][12][13].…”