An operator method for finding exact solutions to vector Korteweg–de Vries equations

Abstract

We develop an operator method which helps finding exact solutions to nonlinear evolution equations (NLEs). Our working schema goes as follows: First we translate the given (NLE) into an appropriate operator version (ONLE). Second, we look for solutions to (ONLE) of the form U=(I+L)−1M, where both L and M are operator-valued functions of the space–time variables and the range of M locates in some appropriate Banach algebras which admits a functional φ that preserves the squares [i.e., φ(A2)=φ(A)2]. Finally, a solution u of the given (NLE) can be obtained by setting u≔φ(U). This method is named by the LM method. Using the LM method, we have rederived the famous Cole–Hopf transformation which reduces the nonlinear Burgers equation into the linear heat equation. The main part of this article is to use the LM method to study the vector Korteweg–de Vries (KdV) equations ut=uxxx+3(u2)x settled in finite-dimensional unital Banach algebras 𝒥. It is shown that these vector KdV equations admit soliton solutions. Specially, we have carried out a thorough study of the quaternionic KdV equation (i.e., the vector KdV equation settled in the Hamilton quaternion algebra ℍ) and shown many interesting and surprising aspects of the quaternionic KdV solitons. Two of them read as follows. (a) The paradoxical energy symmetry breaking phenomenon: Two quaternionic KdV solitons with different energies can annihilate each other. (b) The surprising low-dimensional phenomenon: The interaction of any finitely many quaternionic KdV solitons which live in a unital three-dimensional subspace Π of ℍ does not yield any effect to the part outside that subspace Π and thus their interaction behaves as if it were linear although the interaction between quaternionic KdV solitons is really nonlinear. The LM method can be thought as a complement to the famous bilinear operator method of Hirota. Hirota’s method works very powerful for solving scalar equation but has difficulty with vector equations. The LM method helps overcoming this difficulty.