Fully resonant soliton interactions in the Whitham–Broer–Kaup system based on the double Wronskian solutions

Abstract

With the aim of exploring whether the (1+1)-dimensional coupled nonlinear evolution equations admit abundant soliton interactions, like the cases in the Kadomtsev–Petviashvili II equation, we in this paper study the double Wronskian solutions to the Whitham–Broer–Kaup (WBK) system. We give the parametric condition for two double Wronskians to generate the non-singular, non-trivial and irreducible soliton solutions. Via the asymptotic analysis of two double Wronskians, we show that the soliton solutions of the WBK system is in general linearly combined of fully resonant (M,N)- and (M−1,N+1)-soliton configurations. It turns out that the WBK system can exhibit various complex soliton structures which are different pairwise combinations of elastic, confluent and divergent interactions. From a combinatorial viewpoint, we also explain that the asymptotic solitons of a [(M,N),(M−1,N+1)]-soliton solution are identified by a pair of Grassmannian permutations.

Appendix: Proof of the equivalence relation between Conditions (16) and (17)

We first show the necessity of Condition (17) for the satisfaction of Condition (16). If respectively taking $\mathcal{I}_{1} = \{M+2,M+3,\ldots, L\} \cup \{k\}$, $\mathcal{I}'_{1} = \{M+2, M+3,\ldots, L\}\cup \{k+1\}$ and $\mathcal{I}_{2} = \{M, M+1\ldots,L \} \setminus\{ M+k \}$, $\mathcal{I}'_{2} = \{M, M+1\ldots,L\}\setminus\{ M+k+1 \}$ for any k∈[N−1], from Condition (16) we can obtain the following conditions:

(30)

(31)

which together yield Condition (17).

We next prove the sufficiency of Condition (17) for the satisfaction of Condition (16). Suppose that

$$ \begin{array}{@{}l} \mathcal{I}\setminus \bigl(\mathcal{I}\cap \mathcal{I}'\bigr)= \{i_{k_1}, \ldots, i_{k_{N'} } \}, \\ \noalign {\vspace {5pt}} \mathcal{I}'\setminus \bigl(\mathcal{I}\cap \mathcal{I}'\bigr) = \bigl\{i'_{k_1}, \ldots, i'_{k_{N'} } \bigr\}, \end{array} $$

(32)

where k ₁<⋯<k _N′ and N′≤N. As $\mathcal{I}\cup \mathcal{J}= \mathcal{I}'\cup \mathcal{J}'=[L]$ and $\mathcal{I}\cap \mathcal{J}= \mathcal{I}'\cap \mathcal{J}'= \varnothing$, we then have

$$ \begin{array}{@{}l} \mathcal{J}\setminus \bigl(\mathcal{J}\cap \mathcal{J}'\bigr)= \bigl\{i'_{k_1}, \ldots, i'_{k_{N'}}\bigr\}, \\ \noalign {\vspace {5pt}} \mathcal{J}'\setminus \bigl(\mathcal{J}\cap \mathcal{J}'\bigr)= \{ i_{k_1}, \ldots, i_{k_{N'}}\}. \end{array} $$

(33)

Thus, the left-hand side of Condition (16) can be expressed as

(34)

On the other hand, Condition (17) implies that (−1)^j−i−1 a _i a _j b _i b _j ≤0 for any i,j∈[L]. Accordingly, the sign of the right-hand side of Eq. (34) is determined by

(35)

which implies that Condition (17) is satisfied.