A two-mode coupled Korteweg-de Vries: multiple-soliton solutions and other exact solutions

In this paper, we introduce the new nonlinear two-mode coupled Korteweg-de Vries. We find the necessary conditions of dispersion parameter and the nonlinearity parameter that make this newly coupled give multiple-soliton solutions and multiple singular soliton solutions by using the simplified form of Hirota's direct method. We determine more exact solutions to this new coupled by using other methods such as the sine/cosine method and the sech-expansion method to conduct this study. Finally, all obtained solutions in this paper are new and this coupled is not solved in any other paper.