Using a new technique following the Bogoliubov Mitropolsky method, higher-order nonlinear and dispersive effects are obtained for a kinetic Alfvén wave and ion-acoustic wave with two temperature isothermal electrons, starting from a set of equations that lead to a linear dispersion relation coupling kinetic Alfvén wave ion-acoustic wave. Higher-order solitary wave profiles are obtained for both waves. Graphs are plotted showing the variation of both the Mach number and soliton width against the soliton amplitude for different angles of propagation with the applied magnetic field. Amplitude, width and velocity are found to be greater for larger angles of propagation for both waves. It is found that the higher-order ion-acoustic wave moves faster than the Korteweg–de Vries (KdV) wave, but the higher-order Alfvén wave is slower. The higher-order amplitude and width of both waves are found to be less than those of KdV waves. The potential structure and widths of the double layer for both waves are shown graphically. It is found that the amplitude of the double layer is independent of the angles of propagation, but the widths are greater for larger angles of propagation for both waves.