The nonlinear Schr?dinger(NLS)equation,which incorporates higher-order dispersive terms,is widely employed in the theoretical analysis of various physical phenomena.In this study,we explore the non-commutative extensi...The nonlinear Schr?dinger(NLS)equation,which incorporates higher-order dispersive terms,is widely employed in the theoretical analysis of various physical phenomena.In this study,we explore the non-commutative extension of the higher-order NLS equation.We treat real or complex-valued functions,such as g_(1)=g_(1)(x,t)and g_(2)=g_(2)(x,t)as non-commutative,and employ the Lax pair associated with the evolution equation,as in the commutation case.We derive the quasi-Gramian solution of the system by employing a binary Darboux transformation.The soliton solutions are presented explicitly within the framework of quasideterminants.To visually understand the dynamics and solutions in the given example,we also provide simulations illustrating the associated profiles.Moreover,the solution can be used to study the stability of plane waves and to understand the generation of periodic patterns within the context of modulational instability.展开更多
The effect of temperature variation owing to the cooling pattern （CP） on the microstructural evolution was investigated by establishing a thermomechanical coupled FE （finite element） model. A set of constitutive e...The effect of temperature variation owing to the cooling pattern （CP） on the microstructural evolution was investigated by establishing a thermomechanical coupled FE （finite element） model. A set of constitutive equations of phase transformation was implanted into the commercial FE solver MARC through the user defined subroutine CR- PLAW, and the temperature field was calculated by another user defined subroutine FILM. The results show that the final mierostructure is completely bainite phase for CP one, 98% of bainite phase and 2% of ferrite phase for CP two, and 55% of bainite phase, 35% of pearlite phase and 10% of ferrite phase for CP three.展开更多
基金the support from the National Natural Science Foundation of China,Nos.11835011 and 12375006。
文摘The nonlinear Schr?dinger(NLS)equation,which incorporates higher-order dispersive terms,is widely employed in the theoretical analysis of various physical phenomena.In this study,we explore the non-commutative extension of the higher-order NLS equation.We treat real or complex-valued functions,such as g_(1)=g_(1)(x,t)and g_(2)=g_(2)(x,t)as non-commutative,and employ the Lax pair associated with the evolution equation,as in the commutation case.We derive the quasi-Gramian solution of the system by employing a binary Darboux transformation.The soliton solutions are presented explicitly within the framework of quasideterminants.To visually understand the dynamics and solutions in the given example,we also provide simulations illustrating the associated profiles.Moreover,the solution can be used to study the stability of plane waves and to understand the generation of periodic patterns within the context of modulational instability.
文摘The effect of temperature variation owing to the cooling pattern （CP） on the microstructural evolution was investigated by establishing a thermomechanical coupled FE （finite element） model. A set of constitutive equations of phase transformation was implanted into the commercial FE solver MARC through the user defined subroutine CR- PLAW, and the temperature field was calculated by another user defined subroutine FILM. The results show that the final mierostructure is completely bainite phase for CP one, 98% of bainite phase and 2% of ferrite phase for CP two, and 55% of bainite phase, 35% of pearlite phase and 10% of ferrite phase for CP three.
