In the manuscript, we present several numerical schemes to approximate the coupled nonlinear Schrödinger equations. Three of them are high-order compact and conservative, and the other two are noncompact but conservative. After some numerical analysis, we can find that the schemes are uniquely solvable and convergent. All of them are conservative and stable. By calculating the complexity, we can find that the compact schemes have the same computational cost with the noncompact ones. Numerical illustrations support our analysis. They verify that compact schemes are more efficient than noncompact ones from computation cost and accuracy.

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Background Protocols for Agrobacterium rhizogenes-mediated hairy root transformation of the model legume Lotus japonicus have been established previously. However, little efforts were made in the past to quantify and improve the transformation efficiency. Here, we asked whether effectors (nodulation outer proteins) of the nodule bacterium Sinorhizobium sp. NGR234 can promote hairy root transformation of L. japonicus. The co-expressed red fluorescent protein DsRed1 was used for visualization of transformed roots and for estimation of the transformation efficiency. Results Strong induction of hairy root formation was observed when A. rhizogenes strain LBA9402 was used for L. japonicus transformation. Expression of the effector gene nopP in L. japonicus roots resulted in a significantly increased transformation efficiency while nopL, nopM, and nopT did not show such an effect. In nopP expressing plants, more than 65% of the formed hairy roots were transgenic as analyzed by red fluorescence emitted by co-transformed DsRed1. A nodulation experiment indicated that nopP expression did not obviously affect the symbiosis between L. japonicus and Mesorhizobium loti. Conclusion We have established a novel protocol for hairy root transformation of L. japonicus. The use of A. rhizogenes LBA9402 carrying a binary vector containing DsRed1 and nopP allowed efficient formation and identification of transgenic roots.