Soliton, breather, lump and their interaction solutions of the ( $2+1$ )-dimensional asymmetrical Nizhnik–Novikov–Veselov equation

Advances in Continuous and Discrete Models

Theory and Modern Applications

Table 1 Mixed interaction solutions from five-soliton

N-soliton	Mixed lump-soliton and lump-soliton-breather interaction solutions	Parameters
N = 5	one lump + three solitons	${b_{s}=p_{s} a_{s}}\ (s=1,2,3, 4, 5)$, $a_{1}={l_{1}}\varepsilon $, $a_{2}={l_{2}}\varepsilon$, $a_{3}=a_{4}=\kappa _{1}$, $p_{1}=p_{2}^{} =\gamma _{1}+i\sigma _{1}$, $p_{3}=\gamma _{2}$, $p_{4}=\gamma _{3}$, $a_{5}=\kappa _{2}$, $p_{5}=\gamma _{4}$, $\eta _{01}=\eta _{02}^{}=i\pi $, $\eta _{03}=\eta _{04}=\eta _{05}=0$, ε→0
	one lump + one breather + one soliton	${b_{s}=p_{s} a_{s}}\ (s=1,2,3, 4, 5)$, $a_{1}= {l_{1}}\varepsilon $, $a_{2}={l_{2}}\varepsilon$, $a_{3}=a_{4}=\kappa _{1}$, $p_{1}=p_{2}^{}=\gamma _{1}+i\sigma _{1}$, $p_{3}=p_{4}^{}=\gamma _{2}+i\sigma _{2}$, $a_{5}=\kappa _{2}$, $p_{5}=\gamma _{3}$, $\eta _{01}=\eta _{02}^{*}=i\pi $, $\eta _{03}=\eta _{04}=\eta _{05}=0$, ε→0
	two lumps+one soliton	${b_{s}=p_{s} a_{s}}$, $a_{s}= {l_{s}}\varepsilon\ (s=1,2,3, 4)$, $b_{5}=p_{5} a_{5}$, $p_{1}=p_{2}^{}=\gamma _{1}+i\sigma _{1}$, $p_{3}=p_{4}^{}=\gamma _{2}+i\sigma _{2}$, $a_{5}=\kappa _{2} $, $p_{5}=\gamma _{3}$, $\eta _{01}=\eta _{02}^{}=\eta _{03}=\eta _{04}^{} =i\pi$, $\eta _{05}=0$, ε→0

Note: $\kappa _{1,2}$, $\gamma _{1,2,3,4}$, $\sigma _{1,2}$ are nonzero real constants.

N-soliton	Mixed lump-soliton and lump-soliton-breather interaction solutions	Parameters
N = 5	one lump + three solitons	\({b_{s}=p_{s} a_{s}}\ (s=1,2,3, 4, 5)\), \(a_{1}={l_{1}}\varepsilon \), \(a_{2}={l_{2}}\varepsilon\), \(a_{3}=a_{4}=\kappa _{1}\), \(p_{1}=p_{2}^{} =\gamma _{1}+i\sigma _{1}\), \(p_{3}=\gamma _{2}\), \(p_{4}=\gamma _{3}\), \(a_{5}=\kappa _{2}\), \(p_{5}=\gamma _{4}\), \(\eta _{01}=\eta _{02}^{}=i\pi \), \(\eta _{03}=\eta _{04}=\eta _{05}=0\), ε→0
	one lump + one breather + one soliton	\({b_{s}=p_{s} a_{s}}\ (s=1,2,3, 4, 5)\), \(a_{1}= {l_{1}}\varepsilon \), \(a_{2}={l_{2}}\varepsilon\), \(a_{3}=a_{4}=\kappa _{1}\), \(p_{1}=p_{2}^{}=\gamma _{1}+i\sigma _{1}\), \(p_{3}=p_{4}^{}=\gamma _{2}+i\sigma _{2}\), \(a_{5}=\kappa _{2}\), \(p_{5}=\gamma _{3}\), \(\eta _{01}=\eta _{02}^{*}=i\pi \), \(\eta _{03}=\eta _{04}=\eta _{05}=0\), ε→0
	two lumps+one soliton	\({b_{s}=p_{s} a_{s}}\), \(a_{s}= {l_{s}}\varepsilon\ (s=1,2,3, 4)\), \(b_{5}=p_{5} a_{5}\), \(p_{1}=p_{2}^{}=\gamma _{1}+i\sigma _{1}\), \(p_{3}=p_{4}^{}=\gamma _{2}+i\sigma _{2}\), \(a_{5}=\kappa _{2} \), \(p_{5}=\gamma _{3}\), \(\eta _{01}=\eta _{02}^{}=\eta _{03}=\eta _{04}^{} =i\pi\), \(\eta _{05}=0\), ε→0