The dark Korteweg-de Vries （KdV） systems are defined and classified by Kupershmidt sixteen years ago. However, there is no other classifications for other kinds of nonlinear systems. In this paper, a complete scalar...The dark Korteweg-de Vries （KdV） systems are defined and classified by Kupershmidt sixteen years ago. However, there is no other classifications for other kinds of nonlinear systems. In this paper, a complete scalar classification for dark modified KdV （MKdV） systems is obtained by requiring the existence of higher order differential polynomial symmetries. Different to the nine classes of the dark KdV case, there exist twelve independent classes of the dark MKdV equations. Fhrthermore, for the every class of dark MKdV system, there is a free parameter. Only for a fixed parameter, the dark MKdV can be related to dark KdV via suitable Miura transformation. The reeursion operators of two classes of dark MKdV systems are also given.展开更多
This paper investigates in detail the dynamics of the modified KdV equation with self-consistent sources, including characteristics of one-soliton, scattering conditions and phase shifts of two solitons, degenerate ca...This paper investigates in detail the dynamics of the modified KdV equation with self-consistent sources, including characteristics of one-soliton, scattering conditions and phase shifts of two solitons, degenerate case of two solitons and ＂ghost＂ solitons, etc. Co-moving coordinate frames are employed in asymptotic analysis.展开更多
The dressing method based on the 2×2 matrix■-problem is generalized to study the complex modified KdV equation(cmKdV).Through two linear constraint equations,the spatial and time spectral problems related to the...The dressing method based on the 2×2 matrix■-problem is generalized to study the complex modified KdV equation(cmKdV).Through two linear constraint equations,the spatial and time spectral problems related to the cmKdV equation are derived.The gauge equivalence between the cmKdV equation and the Heisenberg chain equation is obtained.Using a recursive operator,a hierarchy of cmKdV with source is proposed.On the basis of the■-equation,the N-solition solutions of the cmKdV equation are obtained by selecting the appropriate spectral transformation matrix.Furthermore,we get explicit one-soliton and two-soliton solutions.展开更多
The time-fractional modified Korteweg-de Vries(KdV)equation is committed to establish exact solutions by employing the bifurcation method.Firstly,the phase portraits and related qualitative analysis are comprehensivel...The time-fractional modified Korteweg-de Vries(KdV)equation is committed to establish exact solutions by employing the bifurcation method.Firstly,the phase portraits and related qualitative analysis are comprehensively provided.Then,we give parametric expressions of different types of solutions matching with the corresponding orbits.Finally,solution profiles,3D and density plots of some solutions are presented with proper parametric choices.展开更多
In this paper,new doubly-periodic solutions will be found for a new integrable nonlocal modified Korteweg-de Vries equation(mKdV).We adopt a technique using a travelling wave transformation which results in different ...In this paper,new doubly-periodic solutions will be found for a new integrable nonlocal modified Korteweg-de Vries equation(mKdV).We adopt a technique using a travelling wave transformation which results in different solutions represented in Jacobi elliptic functions.展开更多
This paper mainly introduces the parallel physics-informed neural networks(PPINNs)method with regularization strategies to solve the data-driven forward-inverse problems of the variable coefficient modified Korteweg-d...This paper mainly introduces the parallel physics-informed neural networks(PPINNs)method with regularization strategies to solve the data-driven forward-inverse problems of the variable coefficient modified Korteweg-de Vries(VC-MKdV)equation.For the forward problem of the VC-MKdV equation,the authors use the traditional PINN method to obtain satisfactory data-driven soliton solutions and provide a detailed analysis of the impact of network width and depth on solving accuracy and speed.Furthermore,the author finds that the traditional PINN method outperforms the one with locally adaptive activation functions in solving the data-driven forward problems of the VC-MKdV equation.As for the data-driven inverse problem of the VC-MKdV equation,the author introduces a parallel neural networks to separately train the solution function and coefficient function,successfully addressing the function discovery problem of the VC-MKdV equation.To further enhance the network’s generalization ability and noise robustness,the author incorporates two regularization strategies into the PPINNs.An amount of numerical experimental data in this paper demonstrates that the PPINNs method can effectively address the function discovery problem of the VC-MKdV equation,and the inclusion of appropriate regularization strategies in the PPINNs can improves its performance.展开更多
In this study,the generalized modified variable-coefficient KdV equation with external-force term(gvcmKdV)describing atmospheric blocking located in the mid-high latitudes over ocean is studied for integrability prope...In this study,the generalized modified variable-coefficient KdV equation with external-force term(gvcmKdV)describing atmospheric blocking located in the mid-high latitudes over ocean is studied for integrability property by using consistent Riccati expansion solvability and the necessary integrability conditions between the function coefficients are obtained.Moreover,several new solutions have been constructed for the gvcmKdV.Additionally,the classical direct similarity reduction method is used to re-duce the gvcmKdV to a nonlinear ordinary differential equation.Building on the solutions given in the previous literature for the reduced equation,many novel solitary and periodic wave solutions have been obtained for the gvcmKdV.展开更多
In this work we study a new integrable nonlocal modified Korteweg-de Vries equation(mKdV)which arises from a reduction of the AKNS scattering problem.We use a variety of distinct techniques to determine abundant solut...In this work we study a new integrable nonlocal modified Korteweg-de Vries equation(mKdV)which arises from a reduction of the AKNS scattering problem.We use a variety of distinct techniques to determine abundant solutions with distinct physical structures.We show that this nonlocal equation possesses a family of traveling solitary wave solutions that include solitons,kinks,periodic and singular solutions.展开更多
Under investigation in this paper is a complex modified Korteweg–de Vries(KdV) equation, which describes the propagation of short pulses in optical fibers. Bilinear forms and multi-soliton solutions are obtained thro...Under investigation in this paper is a complex modified Korteweg–de Vries(KdV) equation, which describes the propagation of short pulses in optical fibers. Bilinear forms and multi-soliton solutions are obtained through the Hirota method and symbolic computation. Breather-like and bound-state solitons are constructed in which the signs of the imaginary parts of the complex wave numbers and the initial separations of the two parallel solitons are important factors for the interaction patterns. The periodic structures and position-induced phase shift of some solutions are introduced.展开更多
Using the relation between the mKdV equation and the KdV-mKdV equation,we derive non-singular rational solutions for the mKdV equation.The solutions are given in terms of Wronskians.Dynamics of some solutions is inves...Using the relation between the mKdV equation and the KdV-mKdV equation,we derive non-singular rational solutions for the mKdV equation.The solutions are given in terms of Wronskians.Dynamics of some solutions is investigated by means of asymptotic analysis.Wave trajectories of high order rational solutions are asymptotically governed by cubic curves.展开更多
This paper is concerned with a modified transitional Korteweg-de Vries equation ut+f(t)u2ux+uxxx=0, (x,t)∈R+×R+with initial value u(x,0)=g(x)∈H4(R+)and inhomogeneous boundary value u(0,t)=Q(t)∈C2([ 0,∞ )). Un...This paper is concerned with a modified transitional Korteweg-de Vries equation ut+f(t)u2ux+uxxx=0, (x,t)∈R+×R+with initial value u(x,0)=g(x)∈H4(R+)and inhomogeneous boundary value u(0,t)=Q(t)∈C2([ 0,∞ )). Under the conditions either 1) f(t)≤0, f′(t)≥0or 2) f(t)≤−αwhere α>0, we prove the existence of a unique global classical solution.展开更多
We study the following modified transitional Korteweg-de Vries equation ut+f(t)upux+uxxx=0, (x,t)∈R+×R+, (p≥2is an even integer) with initial value u(x,0)=g(x)∈H4(R+)and inhomogeneous boundary value u(0,t)=Q(t...We study the following modified transitional Korteweg-de Vries equation ut+f(t)upux+uxxx=0, (x,t)∈R+×R+, (p≥2is an even integer) with initial value u(x,0)=g(x)∈H4(R+)and inhomogeneous boundary value u(0,t)=Q(t)∈C2([ 0,∞ )). Under the conditions either (i) f(t)≤0, f′(t)≥0or (ii) f(t)≤−αwhere α>0, we prove the existence of a unique global classical solution.展开更多
We analyze a new car-following model described by a differential-difference equation with a synthesized optimal velocity function (SOVF),which depends on the front interactions between every two adjacent vehicles inst...We analyze a new car-following model described by a differential-difference equation with a synthesized optimal velocity function (SOVF),which depends on the front interactions between every two adjacent vehicles instead of the weighted average headway.The model is analyzed with the use of the linear stability theory and nonlinear analysis method.The stability and neutral stability condition are obtained.We also derive the modified KdV (Korteweg-de Vries) equation and the kink-antikink soliton solution near the critical point.A simulation is conducted with integrating the differential-difference equation by the Euler scheme.The results of the numerical simulation verify the validity of the new model.展开更多
By employing the reductive perturbation technique, the propagation of cylindrical and spherical ion acoustic solitary waves is studied in an unmagnetized dense relativistic plasma, consisting of relativistically degen...By employing the reductive perturbation technique, the propagation of cylindrical and spherical ion acoustic solitary waves is studied in an unmagnetized dense relativistic plasma, consisting of relativistically degenerate electrons and cold fluid ions. A modified Korteweg-de-Vries equation is derived and its numerical solutions have been analyzed to identify the basic features of electrostatic solitary structures that may form in such a degenerate Fermi plasma. Different degrees of relativistic electron degeneracy are discussed and compared. It is found that increasing number density leads to decrease the aznplitude the width of the ion acoustic solitary wave in both the cylindrical and spherical geometries. The relevance of the work to the compact astrophysical objects, particularly white dwarfs is pointed out.展开更多
A new two-dimensional lattice hydrodynamic model considering the turning capability of cars is proposed. Based on this model, the stability condition for this new model is obtained by using linear stability analysis. ...A new two-dimensional lattice hydrodynamic model considering the turning capability of cars is proposed. Based on this model, the stability condition for this new model is obtained by using linear stability analysis. Near the critical point, the modified KdV equation is deduced by using the nonlinear theory. The results of numerical simulation indicate that the critical point ac increases with the increase of the fraction p of northbound cars which continue to move along the positive y direction for c = 0.3, but decreases with the increase of p for c = 0.7. The results also indicate that the cars moving along only one direction （eastbound or northbound） are most stable.展开更多
This work explores the construction of more general exact traveling wave solutions of some nonlinear evolution equations(NLEEs)through the application of the(G/G,1/G)-expansion method.This method is allied to the wid...This work explores the construction of more general exact traveling wave solutions of some nonlinear evolution equations(NLEEs)through the application of the(G/G,1/G)-expansion method.This method is allied to the widely used(G/G)-method initiated by Wang et al.and can be considered as an extension of the(G/G)-expansion method.For effectiveness,the method is applied to the family of KdV type equations.Abundant general form solitary wave solutions as well as periodic solutions are successfully obtained through this method.Moreover,in the obtained wider set of solutions,if we set special values of the parameters,some previously known solutions are revived.The approach of this method is simple and elegantly standard.Having been computerized it is also powerful,reliable and effective.展开更多
基金Supported by the Global Change Research Program of China under Grant No.2015Cb953904National Natural Science Foundation of China under Grant Nos.11675054,11435005,11175092,and 11205092+1 种基金Shanghai Knowledge Service Platform for Trustworthy Internet of Things(No.ZF1213)K.C.Wong Magna Fund in Ningbo University
文摘The dark Korteweg-de Vries （KdV） systems are defined and classified by Kupershmidt sixteen years ago. However, there is no other classifications for other kinds of nonlinear systems. In this paper, a complete scalar classification for dark modified KdV （MKdV） systems is obtained by requiring the existence of higher order differential polynomial symmetries. Different to the nine classes of the dark KdV case, there exist twelve independent classes of the dark MKdV equations. Fhrthermore, for the every class of dark MKdV system, there is a free parameter. Only for a fixed parameter, the dark MKdV can be related to dark KdV via suitable Miura transformation. The reeursion operators of two classes of dark MKdV systems are also given.
基金The project supported by National Natural Science Foundation of China under Grant Nos.10371070 and 10671121the Foundation of Shanghai Education Committee for Shanghai Prospective Excellent Young Teachers
文摘This paper investigates in detail the dynamics of the modified KdV equation with self-consistent sources, including characteristics of one-soliton, scattering conditions and phase shifts of two solitons, degenerate case of two solitons and ＂ghost＂ solitons, etc. Co-moving coordinate frames are employed in asymptotic analysis.
基金supported by the National Natural Science Foundation of China(Grant No.12175111,11975131)the KC Wong Magna Fund in Ningbo University.
文摘The dressing method based on the 2×2 matrix■-problem is generalized to study the complex modified KdV equation(cmKdV).Through two linear constraint equations,the spatial and time spectral problems related to the cmKdV equation are derived.The gauge equivalence between the cmKdV equation and the Heisenberg chain equation is obtained.Using a recursive operator,a hierarchy of cmKdV with source is proposed.On the basis of the■-equation,the N-solition solutions of the cmKdV equation are obtained by selecting the appropriate spectral transformation matrix.Furthermore,we get explicit one-soliton and two-soliton solutions.
基金Project supported by the Natural Science Foundation of Shandong Province (Grant No.ZR2021MA084)the Natural Science Foundation of Liaocheng University (Grant No.318012025)Discipline with Strong Characteristics of Liaocheng University–Intelligent Science and Technology (Grant No.319462208)。
文摘The time-fractional modified Korteweg-de Vries(KdV)equation is committed to establish exact solutions by employing the bifurcation method.Firstly,the phase portraits and related qualitative analysis are comprehensively provided.Then,we give parametric expressions of different types of solutions matching with the corresponding orbits.Finally,solution profiles,3D and density plots of some solutions are presented with proper parametric choices.
文摘In this paper,new doubly-periodic solutions will be found for a new integrable nonlocal modified Korteweg-de Vries equation(mKdV).We adopt a technique using a travelling wave transformation which results in different solutions represented in Jacobi elliptic functions.
文摘This paper mainly introduces the parallel physics-informed neural networks(PPINNs)method with regularization strategies to solve the data-driven forward-inverse problems of the variable coefficient modified Korteweg-de Vries(VC-MKdV)equation.For the forward problem of the VC-MKdV equation,the authors use the traditional PINN method to obtain satisfactory data-driven soliton solutions and provide a detailed analysis of the impact of network width and depth on solving accuracy and speed.Furthermore,the author finds that the traditional PINN method outperforms the one with locally adaptive activation functions in solving the data-driven forward problems of the VC-MKdV equation.As for the data-driven inverse problem of the VC-MKdV equation,the author introduces a parallel neural networks to separately train the solution function and coefficient function,successfully addressing the function discovery problem of the VC-MKdV equation.To further enhance the network’s generalization ability and noise robustness,the author incorporates two regularization strategies into the PPINNs.An amount of numerical experimental data in this paper demonstrates that the PPINNs method can effectively address the function discovery problem of the VC-MKdV equation,and the inclusion of appropriate regularization strategies in the PPINNs can improves its performance.
基金The author would like to thank the Deanship of Scientific Re-search,Majmaah University,Saudi Arabia,for funding this work under project No.R-2021-222.
文摘In this study,the generalized modified variable-coefficient KdV equation with external-force term(gvcmKdV)describing atmospheric blocking located in the mid-high latitudes over ocean is studied for integrability property by using consistent Riccati expansion solvability and the necessary integrability conditions between the function coefficients are obtained.Moreover,several new solutions have been constructed for the gvcmKdV.Additionally,the classical direct similarity reduction method is used to re-duce the gvcmKdV to a nonlinear ordinary differential equation.Building on the solutions given in the previous literature for the reduced equation,many novel solitary and periodic wave solutions have been obtained for the gvcmKdV.
文摘In this work we study a new integrable nonlocal modified Korteweg-de Vries equation(mKdV)which arises from a reduction of the AKNS scattering problem.We use a variety of distinct techniques to determine abundant solutions with distinct physical structures.We show that this nonlocal equation possesses a family of traveling solitary wave solutions that include solitons,kinks,periodic and singular solutions.
基金Project supported by the National Natural Science Foundation of China (Grant No. 12161061)the Fundamental Research Funds for the Inner Mongolia University of Finance and Economics (Grant No. NCYWT23036)+2 种基金the Young Innovative and Entrepreneurial Talents of the Inner Mongolia Grassland Talents Project in 2022,Autonomous Region “Five Major Tasks” Research Special Project for the Inner Mongolia University of Finance and Economics in 2024 (Grant No. NCXWD2422)High Quality Research Achievement Cultivation Fund for the Inner Mongolia University of Finance and Economics in 2024 (Grant No. GZCG2426)the Talent Development Fund of Inner Mongolia Autonomous Region, China。
文摘Under investigation in this paper is a complex modified Korteweg–de Vries(KdV) equation, which describes the propagation of short pulses in optical fibers. Bilinear forms and multi-soliton solutions are obtained through the Hirota method and symbolic computation. Breather-like and bound-state solitons are constructed in which the signs of the imaginary parts of the complex wave numbers and the initial separations of the two parallel solitons are important factors for the interaction patterns. The periodic structures and position-induced phase shift of some solutions are introduced.
基金Supported by the National Natural Science Foundation of China under Grant No. 11071157Shanghai Leading Academic Discipline Project under Grant No. J50101
文摘Using the relation between the mKdV equation and the KdV-mKdV equation,we derive non-singular rational solutions for the mKdV equation.The solutions are given in terms of Wronskians.Dynamics of some solutions is investigated by means of asymptotic analysis.Wave trajectories of high order rational solutions are asymptotically governed by cubic curves.
文摘This paper is concerned with a modified transitional Korteweg-de Vries equation ut+f(t)u2ux+uxxx=0, (x,t)∈R+×R+with initial value u(x,0)=g(x)∈H4(R+)and inhomogeneous boundary value u(0,t)=Q(t)∈C2([ 0,∞ )). Under the conditions either 1) f(t)≤0, f′(t)≥0or 2) f(t)≤−αwhere α>0, we prove the existence of a unique global classical solution.
文摘We study the following modified transitional Korteweg-de Vries equation ut+f(t)upux+uxxx=0, (x,t)∈R+×R+, (p≥2is an even integer) with initial value u(x,0)=g(x)∈H4(R+)and inhomogeneous boundary value u(0,t)=Q(t)∈C2([ 0,∞ )). Under the conditions either (i) f(t)≤0, f′(t)≥0or (ii) f(t)≤−αwhere α>0, we prove the existence of a unique global classical solution.
基金supported by National Natural Science Foundation of China under Grant No.60674062the Middle-Aged and Young Scientists Research Incentive Fund of Shandong Province under Grant No.2007BS01013
文摘We analyze a new car-following model described by a differential-difference equation with a synthesized optimal velocity function (SOVF),which depends on the front interactions between every two adjacent vehicles instead of the weighted average headway.The model is analyzed with the use of the linear stability theory and nonlinear analysis method.The stability and neutral stability condition are obtained.We also derive the modified KdV (Korteweg-de Vries) equation and the kink-antikink soliton solution near the critical point.A simulation is conducted with integrating the differential-difference equation by the Euler scheme.The results of the numerical simulation verify the validity of the new model.
基金the Financial Support of HEC Through Indigenous 5000 Ph.D Scholarship Scheme
文摘By employing the reductive perturbation technique, the propagation of cylindrical and spherical ion acoustic solitary waves is studied in an unmagnetized dense relativistic plasma, consisting of relativistically degenerate electrons and cold fluid ions. A modified Korteweg-de-Vries equation is derived and its numerical solutions have been analyzed to identify the basic features of electrostatic solitary structures that may form in such a degenerate Fermi plasma. Different degrees of relativistic electron degeneracy are discussed and compared. It is found that increasing number density leads to decrease the aznplitude the width of the ion acoustic solitary wave in both the cylindrical and spherical geometries. The relevance of the work to the compact astrophysical objects, particularly white dwarfs is pointed out.
基金Project supported by the National Natural Science Foundation of China (Grant Nos. 10865001, 11047003, 11172164, and 11162019)the Youth Foundation of the Yulin Normal University of China (Grant No. 2011YJQN01)
文摘A new two-dimensional lattice hydrodynamic model considering the turning capability of cars is proposed. Based on this model, the stability condition for this new model is obtained by using linear stability analysis. Near the critical point, the modified KdV equation is deduced by using the nonlinear theory. The results of numerical simulation indicate that the critical point ac increases with the increase of the fraction p of northbound cars which continue to move along the positive y direction for c = 0.3, but decreases with the increase of p for c = 0.7. The results also indicate that the cars moving along only one direction （eastbound or northbound） are most stable.
文摘This work explores the construction of more general exact traveling wave solutions of some nonlinear evolution equations(NLEEs)through the application of the(G/G,1/G)-expansion method.This method is allied to the widely used(G/G)-method initiated by Wang et al.and can be considered as an extension of the(G/G)-expansion method.For effectiveness,the method is applied to the family of KdV type equations.Abundant general form solitary wave solutions as well as periodic solutions are successfully obtained through this method.Moreover,in the obtained wider set of solutions,if we set special values of the parameters,some previously known solutions are revived.The approach of this method is simple and elegantly standard.Having been computerized it is also powerful,reliable and effective.
