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The LHCb collaboration reported the observation of a narrow peak in the $D^-K^+$ invariant mass distributions from the $B^+\to D^+ D^- K^+$ decay. The peakis parameterized in terms of two resonances $X_0(2900)$ and $X_1(2900)$ withthe quark contents $\bar{c}\bar{s}ud$, and their spin-parity quantum numbersare $0^+$ and $1^-$, respectively. We investigate the rescattering processeswhich may contribute to the $B^+\to D^+ D^- K^+$ decays. It is shown that the$D^{*-}K^{*+}$ rescattering via the $\chi_{c1}K^{*+}D^{*-}$ loop or the$\bar{D}_{1}^{0}K^{0}$ rescattering via the $D_{sJ}^{+}\bar{D}_{1}^{0}K^{0}$loop simulate the $X_0(2900)$ and $X_1(2900)$ structures. Such phenomena aredue to the analytical property of the scattering amplitudes with the trianglesingularities located to the vicinity of the physical boundary.

Triangle singularity as the origin of $X_0(2900)$ and $X_1(2900)$ observed in $B^+\to D^+ D^- K^+$