A family of wave equations with some remarkable properties

Abstract

We consider a family of homogeneous nonlinear dispersive equations with two arbitrary parameters. Conservation laws are established from the point symmetries and imply that the whole family admits square integrable solutions. Recursion operators are found for two members of the family investigated. For one of them, a Lax pair is also obtained, proving its complete integrability. From the Lax pair, we construct a Miura-type transformation relating the original equation to the Korteweg-de Vries (KdV) equation. This transformation, on the other hand, enables us to obtain solutions of the equation from the kernel of a Schrödinger operator with potential parametrized by the solutions of the KdV equation. In particular, this allows us to exhibit a kink solution to the completely integrable equation from the 1-soliton solution of the KdV equation. Finally, peakon-type solutions are also found for a certain choice of the parameters, although for this particular case the equation is reduced to a homogeneous second-order nonlinear evolution equation.

Kink-type solutions of the SIdV equation and their properties

“…The trajectories, the phase shifts, and the decomposition of the first three kink-type solutions u [ n ] ( n = 1, 2, 3) are studied in detail. A crucial relationship is u = ψ | λ =0 , so we can use the DT to construct the solution u [ n ] without using the solution of the associated Legendre equation as was done in [2]. By comparing with the results reported in [2], we believe that our method presented here is simpler and systematic.…”

“…A crucial relationship is u = ψ | λ =0 , so we can use the DT to construct the solution u [ n ] without using the solution of the associated Legendre equation as was done in [2]. By comparing with the results reported in [2], we believe that our method presented here is simpler and systematic. Moreover, we mention that the SIdV equation is also used to describe and control the revolution of surfaces [7,8], thus it is an interesting issue to get explicitly the surfaces associated with the order- n kink soliton u [ n ] .…”

“…Very recently, Silva et al . [2] have put forward the connection between SIdV, Airy (linear KdV), KdV and modified KdV equations. We set ϵa = 1 and ϵ = 2/3, then equation (1.3) becomes [1,2]ut+3uxuxxu=uxxx,which admits the following Lax pair:L=−Dx2+uxxu1emand1emB=4Dx3−6uxxuDx−3(uxxxu−uxuxxu2).Note that the SIdV equation (1.4) was deduced as early as almost 30 years ago in [3] from a point of view of the governing equation of eigenfunction by eliminating the potential function ω in Lax pair, and this equation was also re-derived in [4] by revealing the direct links between the well-known Sylvester matrix equation and soliton equations.…”

“…If ω is a solution of the KdV equation (1.2), then a solution of the SIdV equation (1.4) can be obtained by solving the following linear system [2]:ut+3ωux=uxxx2emand2em−uxx+ωu=0.}It is highly non-trivial to get the solution u of the SIdV equation (1.4) from a known solution ω of the KdV equation from the above coupled linear system of variable coefficient partial differential equations. Silva et al .…”

“…Silva et al . [2] have obtained a kink-type solution u from a single soliton solution ω by solving the associated Legendre equation that is reduced from the second formula of equation (1.6). Obviously, the idea to get the solution u from the solution ω through solving the linear system (1.6) might be not applicable if the known solution ω is a two-soliton solution or other complicated solution of the KdV equation.…”