In solid physics equations, accurate and novel soliton wave structures for heating a single crystal of sodium fluoride.

This paper analyzes the analytical and numerical solutions' structure of the combined mKdV equation and KdV equation (mKdV+KdV equation) using the Khater II (Khat. II) method and three accurate B-spline numerical schemes. ExCBS, SBS and TQBS numerical schemes are the numerical systems used. The handled model describes many distinct phenomena such as wave propagation of bounded particles with a harmonic force in a one-dimensional nonlinear lattice, propagation of ion-acoustic waves of small amplitude without Landau damping in plasma physics, and propagation of thermal pulse through a single sodium fluoride crystal in solid physics. Numerous examples show the relationship between quick and slow soliton, which generates phase shift. This phase shift is shown in a contour map to show the modest and colossal energy density along the path of fast and slow colliding solitons. Calculating the difference between analytical and numerical solutions shows whether they match spline-connected and distribution graphs. [ABSTRACT FROM AUTHOR]