数学归纳法是数学中常用的一种证题方法.这种方法在形式上有许多不同的模式,通常我们采用的“1对;假设 K 对;那末 K+1也对”是最简单的一种.应用数学归纳法时,如果只注意形式的照搬硬套,而忽视归纳法三要点(起点、假设和递推)之间的协...数学归纳法是数学中常用的一种证题方法.这种方法在形式上有许多不同的模式,通常我们采用的“1对;假设 K 对;那末 K+1也对”是最简单的一种.应用数学归纳法时,如果只注意形式的照搬硬套,而忽视归纳法三要点(起点、假设和递推)之间的协调一致关系,往往会变成一种形式上象数学归纳法而实质上却不成其为数学归纳法的错误证法.这种情况,在二元命题的证明中不但容易发生。展开更多
Under consideration in this study is the discrete coupled modified Korteweg-de Vries(mKdV)equation with 4×4 Lax pair.Firstly,through using continuous limit technique,this discrete equation can be mapped to the co...Under consideration in this study is the discrete coupled modified Korteweg-de Vries(mKdV)equation with 4×4 Lax pair.Firstly,through using continuous limit technique,this discrete equation can be mapped to the coupled KdV and mKdV equations,which may depict the development of shallow water waves,the optical soliton propagation in cubic nonlinear media and the Alfven wave in a cold collision-free plasma.Secondly,the discrete generalized(r,N-r)-fold Darboux transformation is constructed and extended to solve this discrete coupled equation with the fourth-order linear spectral problem,from which diverse exact solutions including usual multi-soliton and semi-rational soliton solutions on the vanishing background,higher-order rational soliton and mixed hyperbolic-rational soliton solutions on the non-vanishing background are derived,and the limit states of some soliton and rational soliton solutions are analyzed by the asymptotic analysis technique.Finally,the numerical simulations are used to explore the dynamical behaviors of some exact soliton solutions.These results may be helpful for understanding some physical phenomena in fields of shallow water wave,optics,and plasma physics.展开更多
文摘数学归纳法是数学中常用的一种证题方法.这种方法在形式上有许多不同的模式,通常我们采用的“1对;假设 K 对;那末 K+1也对”是最简单的一种.应用数学归纳法时,如果只注意形式的照搬硬套,而忽视归纳法三要点(起点、假设和递推)之间的协调一致关系,往往会变成一种形式上象数学归纳法而实质上却不成其为数学归纳法的错误证法.这种情况,在二元命题的证明中不但容易发生。
基金Supported by the National Natural Science Foundation of China(11871452,12071052the Natural Science Foundation of Henan(202300410338)the Nanhu Scholar Program for Young Scholars of XYNU。
基金Project supported by the National Natural Science Foundation of China (Grant No.12071042)Beijing Natural Science Foundation (Grant No.1202006)。
文摘Under consideration in this study is the discrete coupled modified Korteweg-de Vries(mKdV)equation with 4×4 Lax pair.Firstly,through using continuous limit technique,this discrete equation can be mapped to the coupled KdV and mKdV equations,which may depict the development of shallow water waves,the optical soliton propagation in cubic nonlinear media and the Alfven wave in a cold collision-free plasma.Secondly,the discrete generalized(r,N-r)-fold Darboux transformation is constructed and extended to solve this discrete coupled equation with the fourth-order linear spectral problem,from which diverse exact solutions including usual multi-soliton and semi-rational soliton solutions on the vanishing background,higher-order rational soliton and mixed hyperbolic-rational soliton solutions on the non-vanishing background are derived,and the limit states of some soliton and rational soliton solutions are analyzed by the asymptotic analysis technique.Finally,the numerical simulations are used to explore the dynamical behaviors of some exact soliton solutions.These results may be helpful for understanding some physical phenomena in fields of shallow water wave,optics,and plasma physics.
