Nonlocal symmetries and molecule structures of the KdV hierarchy

Abstract

The nonlocal symmetries are important because they carry information about the existence of Darboux–Bäcklund and linearizing transformations, and they also allow us to construct nontrivial solutions to systems. In this paper, we focus on investigating three sets of nonlocal symmetries which are realized as appropriate local symmetries of related auxiliary systems for the Korteweg–de Vries hierarchy. Specifically, we construct infinitely many nonlocal symmetries from three different aspects using residues, Lax pairs and quadratic pseudopotentials for each member of the Korteweg–de Vries hierarchy and show how these nonlocal symmetries connect each other. These infinitely many nonlocal symmetries enable us to construct multiple soliton solutions. Moreover, we adapt the multiple soliton solutions derived from nonlocal symmetries and describe a procedure by using velocity resonance mechanism to find molecule solutions, which can be found not only in the optical systems, but also in fluid systems, for the combined third-fifth-order Korteweg–de Vries equation. Several types of molecule structures including soliton molecules, breather molecules and soliton–breather molecules are illustrated.

Appendix A

Proposition 1

The Lax pair for the nth equation of the KdV hierarchy is

$$\begin{aligned}&\psi _{xx}-(\lambda -u)\psi =0,\\&\psi _{t_{n+1}}=\frac{T_x(u,\lambda )}{2}\psi -T(u,\lambda )\psi _x \end{aligned}$$

with

$$\begin{aligned} T(u,\lambda )=-2\sum _{k=0}^{n}((4\lambda )^{n-k}L_k[u]). \end{aligned}$$

Proposition 2

The nth equation of the KdV hierarchy admits $\phi$ determined by the compatible equations

$$\begin{aligned}&\phi _{x}=\psi ^2, \end{aligned}$$

(50)

$$\begin{aligned}&\phi _{t_{n+1}}=Q(u,\lambda )\psi ^2+R(u,\lambda )\psi _x^2-R_x(u,\lambda )\psi \psi _x \end{aligned}$$

(51)

with

$$\begin{aligned}&R(u,\lambda )=-8\sum _{k=0}^{n-1}((n-k)(4\lambda )^{n-k-1}L_{k}[u]),\\&Q(u,\lambda )=\left( \frac{\partial _{xx}}{2}+u-\lambda \right) R(u,\lambda )-T(u,\lambda ). \end{aligned}$$

Proof

The compatibility condition of Eqs. (50) and (51) implies

$$\begin{aligned}&(T_x(u,\lambda )-Q_x(u,\lambda )-(u-\lambda )R_x(u,\lambda ))\psi ^2\\&\quad +(R_{xx}(u,\lambda )+2(u-\lambda )R(u,\lambda )\\&\quad -2T(u,\lambda )-2Q(u,\lambda ))\psi \psi _x=0. \end{aligned}$$

Setting coefficients of $\psi ^2$ and $\psi \psi _x$ to zero, we get

$$\begin{aligned}&\psi \psi _x: Q(u,\lambda )=\frac{R_{xx}(u,\lambda )}{2}\nonumber \\&\quad +(u-\lambda )R(u,\lambda )-T(u,\lambda ),\end{aligned}$$

(52)

$$\begin{aligned}&\psi ^2: (\partial _{xxx}+2u_x+4(u-\lambda )\partial _x)R(u,\lambda )\nonumber \\&\quad =4T_x(u,\lambda ). \end{aligned}$$

(53)

Substituting

$$\begin{aligned}&Q(u,\lambda )=\sum _{i=0}^{n}q_i[u]\lambda ^i,\quad R(u,\lambda )=\sum _{i=0}^{n}r_i[u]\lambda ^i,\\&T(u,\lambda )=-2\sum _{k=0}^{n}((4\lambda )^{n-k}L_k[u]) \end{aligned}$$

into Eqs. (52) and (53), one can show by straightforward computations that

$$\begin{aligned}&R(u,\lambda )=-8\sum _{k=0}^{n-1}((n-k)(4\lambda )^{n-k-1}L_{k}[u]),\\&Q(u,\lambda )=\left( \frac{\partial _{xx}}{2}+u-\lambda \right) R(u,\lambda )-T(u,\lambda ). \end{aligned}$$

$\square$