We study the complex Sharma-Tasso-Olver equation using the Riemann-Hilbert approach.The associated Riemann-Hilbert problem for this integrable equation can be naturally constructed by considering the spectral problem ...We study the complex Sharma-Tasso-Olver equation using the Riemann-Hilbert approach.The associated Riemann-Hilbert problem for this integrable equation can be naturally constructed by considering the spectral problem of the Lax pair.Subsequently,in the case that the Riemann-Hilbert problem is irregular,the N-soliton solutions of the equation can be deduced.In addition,the three-dimensional graphic of the soliton solutions and wave propagation image are graphically depicted and further discussed.展开更多
In this work,using the Hirota bilinear method,N-soliton solution is obtained for Hirota-Satsuma nonlinear evolution equation:u_t - u_(xxt) - 3u_xu_t + u_x = 0.
By means of Hirota method,N-soliton solutions of the modified KdV equation under the Bargmannconstraint are obtained through solving the Bargmann constraint and the related Lax pair and conjugate Lax pair ofthe modifi...By means of Hirota method,N-soliton solutions of the modified KdV equation under the Bargmannconstraint are obtained through solving the Bargmann constraint and the related Lax pair and conjugate Lax pair ofthe modified KdV equation.展开更多
The bilinear form of the （2＋1）-dimensional non-isospectral AKNS system is derived. Its N-soliton solutions are obtained by using the Hirota method. As a reduction, a （2＋1）-dimensional non-isospectral Schrodinger...The bilinear form of the （2＋1）-dimensional non-isospectral AKNS system is derived. Its N-soliton solutions are obtained by using the Hirota method. As a reduction, a （2＋1）-dimensional non-isospectral Schrodinger equation and its N-soliton solutions are constructed.展开更多
Based on the Hirota bilinear form, a simple approach without employing the standard perturbation technique, is presented for constructing a novel N-soliton solution for a （3＋1）-dimensional nonlinear evolution equat...Based on the Hirota bilinear form, a simple approach without employing the standard perturbation technique, is presented for constructing a novel N-soliton solution for a （3＋1）-dimensional nonlinear evolution equation. Moreover, the novel N-soliton solution is shown to have resonant behavior with the aid of Mathematica.展开更多
In this paper, a nonlocal two-wave interaction system from the Manakov hierarchy is investigated via the Riemann–Hilbert approach. Based on the spectral analysis of the Lax pair, a Riemann–Hilbert problem for the no...In this paper, a nonlocal two-wave interaction system from the Manakov hierarchy is investigated via the Riemann–Hilbert approach. Based on the spectral analysis of the Lax pair, a Riemann–Hilbert problem for the nonlocal two-wave interaction system is constructed. By discussing the solutions of this Riemann–Hilbert problem in both the regular and nonregular cases, we explicitly present the N-soliton solution formula of the nonlocal two-wave interaction system. Moreover,the dynamical behaviour of the single-soliton solution is shown graphically.展开更多
Using the Hirota's bilinear method,some new N-soliton solution are presented for two multidimensional analogues of the m-KdV equation wt+wxxx-6w 2 wx+3 2( w x -1 wy+w-x -1 wz)x=0 and wt+wxxx?6w 2 wx+3 2( wwy+wx-x-...Using the Hirota's bilinear method,some new N-soliton solution are presented for two multidimensional analogues of the m-KdV equation wt+wxxx-6w 2 wx+3 2( w x -1 wy+w-x -1 wz)x=0 and wt+wxxx?6w 2 wx+3 2( wwy+wx-x-1 wy)=0 in view of a different treatment.展开更多
The general bright-dark mixed N-soliton solution of the two-dimensional Maccari system is obtained with the KP hierarchy reduction method. The dynamics of single and two solitons are discussed in detail. Asymptotic an...The general bright-dark mixed N-soliton solution of the two-dimensional Maccari system is obtained with the KP hierarchy reduction method. The dynamics of single and two solitons are discussed in detail. Asymptotic analysis shows that two solitons undergo elastic collision accompanied by a position shift. Furthermore, our analysis on mixed soliton bound states shows that arbitrary higher-order soliton bound states can take place.展开更多
An explicit N-fold Darboux transformation with multiparameters for nonlinear Schrodinger equation is constructed with the help of its Lax pairs and a reduction technique. According to this Darboux transformation, the ...An explicit N-fold Darboux transformation with multiparameters for nonlinear Schrodinger equation is constructed with the help of its Lax pairs and a reduction technique. According to this Darboux transformation, the solutions of the nonlinear Schrfdinger equation are reduced to solving a linear algebraic system, from which a unified and explicit formulation of N-soliton solutions with multiparameters for the nonlinear Schrfdinger equation is given.展开更多
In this paper, we studied N-soliton solutions of a new integrable equation studied by Qiao [J. Math. Phys. 48 082701 （2007）]. Firstly, we employed the Darboux matrix method to construct a Darboux transformation for ...In this paper, we studied N-soliton solutions of a new integrable equation studied by Qiao [J. Math. Phys. 48 082701 （2007）]. Firstly, we employed the Darboux matrix method to construct a Darboux transformation for the modified Korteweg-de Vries equation. Then we use the Darboux transformation and a transformation, introduced by Sakovich [J. Math. Phys. 52 023509 （2011）], to derive N-soliton solutions of the new integrable equation from the seed solution. In particular, the multiple soliton solutions are explicitly obtained and shown through some figures.展开更多
These rational solutions which can be described a kind of algebraic solitary waves which have great potential in applied value in atmosphere and ocean. It has attracted more and more attention recently. In this paper,...These rational solutions which can be described a kind of algebraic solitary waves which have great potential in applied value in atmosphere and ocean. It has attracted more and more attention recently. In this paper, the generalized bilinear method instead of the Hirota bilinear method is used to obtain the rational solutions to the (2 + 1)-dimensional Boiti-Leon-Manna-Pempinelli-like equation (hereinafter referred to as BLMP equation). Meanwhile, the (2 + 1)-dimensional BLMP-like equation is derived on the basis of the generalized bilinear operators D3,x D3,y and D3,t. And the rational solutions to the (2 + 1)-dimensional BLMP-like equation are obtained successively. Finally, with the help of the N-soliton solutions of the (2 + 1)-dimensional BLMP equation, the interactions of the N-soliton solutions can be derived. The results show that the two soliton still maintained the original waveform after happened collision.展开更多
A new coupled integrable dispersionless equation is presented by considering a spectral problem. A Darboux transformation for the resulting coupled integrable dispersionless equation is constructed with the help of sp...A new coupled integrable dispersionless equation is presented by considering a spectral problem. A Darboux transformation for the resulting coupled integrable dispersionless equation is constructed with the help of spectral problems. As an application, the N-soliton solution of the coupled integrable dispersionless equation is explicitly given.展开更多
In this article, we focus on investigating the Kundu-type equation with zero boundary condition at infinity. Based on the analytical and symmetric properties of eigenfunctions and spectral matrix of its Lax pair, a Ri...In this article, we focus on investigating the Kundu-type equation with zero boundary condition at infinity. Based on the analytical and symmetric properties of eigenfunctions and spectral matrix of its Lax pair, a Riemann-Hilbert problem for the initial value problem of the Kundu-type equation is constructed. Further through solving the regular and nonregular Riemann-Hilbert problem, a kind of general N-soliton solution of the Kundu-type equation are presented. As special cases of this result, the N-soliton solution of the Kaup-Newell equation, Chen-Lee-Liu equation, and Gerjikov-Ivanov equation can be obtained respectively by choosing different parameters.展开更多
We present new lemmas,theorem and corollaries to construct interactions among higher-order rogue waves,n-periodic waves and n-solitons solutions(n→∞)to the(2+1)-dimensional asymmetric Nizhnik-Novikov-Veselov(ANNV)eq...We present new lemmas,theorem and corollaries to construct interactions among higher-order rogue waves,n-periodic waves and n-solitons solutions(n→∞)to the(2+1)-dimensional asymmetric Nizhnik-Novikov-Veselov(ANNV)equation.Several examples for theories are given by choosing definite interactions of the wave solutions for the model.In particular,we exhibit dynamical interactions between a rogue and a cross bright-dark bell wave,a rogue and a cross-bright bell wave,a rogue and a one-,two-,three-,four-periodic wave.In addition,we also present multi-types interactions between a rogue and a periodic cross-bright bell wave,a rogue and a periodic cross-bright-bark bell wave.Finally,we physically explain such interaction solutions of the model in the 3D and density plots.展开更多
By employing the Hirota’s bilinear method and different test functions, the breather solutions of HSI equation with different structures are obtained based on symbolic calculation with perturbation parameters. Some n...By employing the Hirota’s bilinear method and different test functions, the breather solutions of HSI equation with different structures are obtained based on symbolic calculation with perturbation parameters. Some new lump solitons are found in the process of studying the degradation behavior of breather solutions. The interaction between lump solution and soliton solution is constructed in the form of lump solution, and the motion trajectory of lump is obtained. In addition, the theorem of lump solitons and N-solitons superposition is given and proved. The superposition formula of lump is derived from the theorem, and its spatial evolution behavior is given.展开更多
In this paper,the N-soliton solutions to the nonlocal reverse space-time Chen-Lee-Liu equation have been derived.Under the nonlocal symmetry reduction to the matrix spectral problem,the nonlocal reverse space-time Che...In this paper,the N-soliton solutions to the nonlocal reverse space-time Chen-Lee-Liu equation have been derived.Under the nonlocal symmetry reduction to the matrix spectral problem,the nonlocal reverse space-time Chen-Lee-Liu equation can be obtained.Based on the spectral problem,the specific matrix Riemann-Hilbert problem is constructed for this nonlocal equation.Through solving this associated Riemann-Hilbert problem,the N-soliton solutions to this nonlocal equation can be obtained in the case of the jump matrix as an identity matrix.展开更多
This paper is concerned with the Navier-Stokes/Allen-Cahn system,which is used to model the dynamics of immiscible two-phase flows.We consider a 1D free boundary problem and assume that the viscosity coefficient depen...This paper is concerned with the Navier-Stokes/Allen-Cahn system,which is used to model the dynamics of immiscible two-phase flows.We consider a 1D free boundary problem and assume that the viscosity coefficient depends on the density in the form ofη(ρ)=ρ^(α).The existence of unique global H^(2m)-solutions(m∈N)to the free boundary problem is proven for when 0<α<1/4.Furthermore,we obtain the global C^(∞)-solutions if the initial data is smooth.展开更多
Solid solution-strengthened copper alloys have the advantages of a simple composition and manufacturing process,high mechanical and electrical comprehensive performances,and low cost;thus,they are widely used in high-...Solid solution-strengthened copper alloys have the advantages of a simple composition and manufacturing process,high mechanical and electrical comprehensive performances,and low cost;thus,they are widely used in high-speed rail contact wires,electronic component connectors,and other devices.Overcoming the contradiction between low alloying and high performance is an important challenge in the development of solid solution-strengthened copper alloys.Taking the typical solid solution-strengthened alloy Cu-4Zn-1Sn as the research object,we proposed using the element In to replace Zn and Sn to achieve low alloying in this work.Two new alloys,Cu-1.5Zn-1Sn-0.4In and Cu-1.5Zn-0.9Sn-0.6In,were designed and prepared.The total weight percentage content of alloying elements decreased by 43%and 41%,respectively,while the product of ultimate tensile strength(UTS)and electrical conductivity(EC)of the annealed state increased by 14%and 15%.After cold rolling with a 90%reduction,the UTS of the two new alloys reached 576 and 627MPa,respectively,the EC was 44.9%IACS and 42.0%IACS,and the product of UTS and EC(UTS×EC)was 97%and 99%higher than that of the annealed state alloy.The dislocations proliferated greatly in cold-rolled alloys,and the strengthening effects of dislocations reached 332 and 356 MPa,respectively,which is the main reason for the considerable improvement in mechanical properties.展开更多
This paper studies the(2+1)-dimensional Hirota-Satsuma-Ito equation.Based on an associated Hirota bilinear form,lump-type solution,two types of interaction solutions,and breather wave solution of the(2+1)-dimensional ...This paper studies the(2+1)-dimensional Hirota-Satsuma-Ito equation.Based on an associated Hirota bilinear form,lump-type solution,two types of interaction solutions,and breather wave solution of the(2+1)-dimensional Hirota-Satsuma-Ito equation are obtained,which are all related to the seed solution of the equation.It is interesting that the rogue wave is aroused by the interaction between one-lump soliton and a pair of resonance stripe solitons,and the fusion and fission phenomena are also found in the interaction between lump solitons and one-stripe soliton.Furthermore,the breather wave solution is also obtained by reducing the two-soliton solutions.The trajectory and period of the one-order breather wave are analyzed.The corresponding dynamical characteristics are demonstrated by the graphs.展开更多
A numerical approach is an effective means of solving boundary value problems(BVPs).This study focuses on physical problems with general partial differential equations(PDEs).It investigates the solution approach throu...A numerical approach is an effective means of solving boundary value problems(BVPs).This study focuses on physical problems with general partial differential equations(PDEs).It investigates the solution approach through the standard forms of the PDE module in COMSOL.Two typical mechanics problems are exemplified:The deflection of a thin plate,which can be addressed with the dedicated finite element module,and the stress of a pure bending beamthat cannot be tackled.The procedure for the two problems regarding the three standard forms required by the PDE module is detailed.The results were in good agreement with the literature,indicating that the PDE module provides a promising means to solve complex PDEs,especially for those a dedicated finite element module has yet to be developed.展开更多
基金Project supported by the National Natural Science Foundation of China(Grant No.11975145)the Program for Science&Technology Innovation Talents in Universities of Henan Province,China(Grant No.22HASTIT019)+2 种基金the Natural Science Foundation of Henan,China(Grant No.202300410524)the Science and Technique Project of Henan,China(Grant No.212102310397)the Academic Degrees&Graduate Education Reform Project of Henan Province,China(Grant No.2021SJGLX219Y)。
文摘We study the complex Sharma-Tasso-Olver equation using the Riemann-Hilbert approach.The associated Riemann-Hilbert problem for this integrable equation can be naturally constructed by considering the spectral problem of the Lax pair.Subsequently,in the case that the Riemann-Hilbert problem is irregular,the N-soliton solutions of the equation can be deduced.In addition,the three-dimensional graphic of the soliton solutions and wave propagation image are graphically depicted and further discussed.
基金Foundation item： Supported by the Natural Science Foundation of China（61072147, 11071159） Supported by the Shanghai Leading Academic Discipline Project（J50101） Supported by the Youth Foundation of Zhoukou Normal University（zknuqn200917）
文摘In this work,using the Hirota bilinear method,N-soliton solution is obtained for Hirota-Satsuma nonlinear evolution equation:u_t - u_(xxt) - 3u_xu_t + u_x = 0.
基金Supported by the National Natural Science Foundation of China under Grant Nos.10371070 and 10671121
文摘By means of Hirota method,N-soliton solutions of the modified KdV equation under the Bargmannconstraint are obtained through solving the Bargmann constraint and the related Lax pair and conjugate Lax pair ofthe modified KdV equation.
基金supported by China Postdoctoral Science Foundation and National Natural Science Foundation of China under Grant No.10771207
文摘The bilinear form of the （2＋1）-dimensional non-isospectral AKNS system is derived. Its N-soliton solutions are obtained by using the Hirota method. As a reduction, a （2＋1）-dimensional non-isospectral Schrodinger equation and its N-soliton solutions are constructed.
文摘Based on the Hirota bilinear form, a simple approach without employing the standard perturbation technique, is presented for constructing a novel N-soliton solution for a （3＋1）-dimensional nonlinear evolution equation. Moreover, the novel N-soliton solution is shown to have resonant behavior with the aid of Mathematica.
基金Project supported by the National Natural Science Foundation of China(Grant Nos.11331008 and 11522112)
文摘In this paper, a nonlocal two-wave interaction system from the Manakov hierarchy is investigated via the Riemann–Hilbert approach. Based on the spectral analysis of the Lax pair, a Riemann–Hilbert problem for the nonlocal two-wave interaction system is constructed. By discussing the solutions of this Riemann–Hilbert problem in both the regular and nonregular cases, we explicitly present the N-soliton solution formula of the nonlocal two-wave interaction system. Moreover,the dynamical behaviour of the single-soliton solution is shown graphically.
基金Supported by the National Natural Science Foundation of China（10871132 11074160） Supported by the National Natura Science Foundation of Henan Province（102300410190 092300410202）
文摘Using the Hirota's bilinear method,some new N-soliton solution are presented for two multidimensional analogues of the m-KdV equation wt+wxxx-6w 2 wx+3 2( w x -1 wy+w-x -1 wz)x=0 and wt+wxxx?6w 2 wx+3 2( wwy+wx-x-1 wy)=0 in view of a different treatment.
基金Supported by the Global Change Research Program of China under Grant No 2015CB953904the National Natural Science Foundation of China under Grant Nos 11675054 and 11435005the Shanghai Collaborative Innovation Center of Trustworthy Software for Internet of Things under Grant No ZF1213
文摘The general bright-dark mixed N-soliton solution of the two-dimensional Maccari system is obtained with the KP hierarchy reduction method. The dynamics of single and two solitons are discussed in detail. Asymptotic analysis shows that two solitons undergo elastic collision accompanied by a position shift. Furthermore, our analysis on mixed soliton bound states shows that arbitrary higher-order soliton bound states can take place.
文摘An explicit N-fold Darboux transformation with multiparameters for nonlinear Schrodinger equation is constructed with the help of its Lax pairs and a reduction technique. According to this Darboux transformation, the solutions of the nonlinear Schrfdinger equation are reduced to solving a linear algebraic system, from which a unified and explicit formulation of N-soliton solutions with multiparameters for the nonlinear Schrfdinger equation is given.
基金Project supported by the National Natural Science Foundation of China (Grant No. 11261037)the High Education Science Research Fund of China (Grant No. 211034)the High Education Science Research Program of Inner Mongolia Autonomous Region, China (Grant No. NJ10045)
文摘In this paper, we studied N-soliton solutions of a new integrable equation studied by Qiao [J. Math. Phys. 48 082701 （2007）]. Firstly, we employed the Darboux matrix method to construct a Darboux transformation for the modified Korteweg-de Vries equation. Then we use the Darboux transformation and a transformation, introduced by Sakovich [J. Math. Phys. 52 023509 （2011）], to derive N-soliton solutions of the new integrable equation from the seed solution. In particular, the multiple soliton solutions are explicitly obtained and shown through some figures.
文摘These rational solutions which can be described a kind of algebraic solitary waves which have great potential in applied value in atmosphere and ocean. It has attracted more and more attention recently. In this paper, the generalized bilinear method instead of the Hirota bilinear method is used to obtain the rational solutions to the (2 + 1)-dimensional Boiti-Leon-Manna-Pempinelli-like equation (hereinafter referred to as BLMP equation). Meanwhile, the (2 + 1)-dimensional BLMP-like equation is derived on the basis of the generalized bilinear operators D3,x D3,y and D3,t. And the rational solutions to the (2 + 1)-dimensional BLMP-like equation are obtained successively. Finally, with the help of the N-soliton solutions of the (2 + 1)-dimensional BLMP equation, the interactions of the N-soliton solutions can be derived. The results show that the two soliton still maintained the original waveform after happened collision.
基金supported by the National Key Basic Research Project of China(Grant No 2004CB318000)the National Natural Science Foundation of China(Grant No 10771072)+2 种基金the Research Fund far the Doctoral Program of Higher Education of China(Grant No 20060269006)the PhD Program Scholarship Fund of East China Normal University(ECNU)2008,China(Grant No 20080052)the Natural Science Foundation of Inner Mongolia Normal University,China(Grant No ZRYB08017)
文摘A new coupled integrable dispersionless equation is presented by considering a spectral problem. A Darboux transformation for the resulting coupled integrable dispersionless equation is constructed with the help of spectral problems. As an application, the N-soliton solution of the coupled integrable dispersionless equation is explicitly given.
基金supported by the National Science Foundation of China(11671095,51879045,11805114)。
文摘In this article, we focus on investigating the Kundu-type equation with zero boundary condition at infinity. Based on the analytical and symmetric properties of eigenfunctions and spectral matrix of its Lax pair, a Riemann-Hilbert problem for the initial value problem of the Kundu-type equation is constructed. Further through solving the regular and nonregular Riemann-Hilbert problem, a kind of general N-soliton solution of the Kundu-type equation are presented. As special cases of this result, the N-soliton solution of the Kaup-Newell equation, Chen-Lee-Liu equation, and Gerjikov-Ivanov equation can be obtained respectively by choosing different parameters.
文摘We present new lemmas,theorem and corollaries to construct interactions among higher-order rogue waves,n-periodic waves and n-solitons solutions(n→∞)to the(2+1)-dimensional asymmetric Nizhnik-Novikov-Veselov(ANNV)equation.Several examples for theories are given by choosing definite interactions of the wave solutions for the model.In particular,we exhibit dynamical interactions between a rogue and a cross bright-dark bell wave,a rogue and a cross-bright bell wave,a rogue and a one-,two-,three-,four-periodic wave.In addition,we also present multi-types interactions between a rogue and a periodic cross-bright bell wave,a rogue and a periodic cross-bright-bark bell wave.Finally,we physically explain such interaction solutions of the model in the 3D and density plots.
文摘By employing the Hirota’s bilinear method and different test functions, the breather solutions of HSI equation with different structures are obtained based on symbolic calculation with perturbation parameters. Some new lump solitons are found in the process of studying the degradation behavior of breather solutions. The interaction between lump solution and soliton solution is constructed in the form of lump solution, and the motion trajectory of lump is obtained. In addition, the theorem of lump solitons and N-solitons superposition is given and proved. The superposition formula of lump is derived from the theorem, and its spatial evolution behavior is given.
基金supported by the National Natural Science Foundation of China under Grant No.11975145。
文摘In this paper,the N-soliton solutions to the nonlocal reverse space-time Chen-Lee-Liu equation have been derived.Under the nonlocal symmetry reduction to the matrix spectral problem,the nonlocal reverse space-time Chen-Lee-Liu equation can be obtained.Based on the spectral problem,the specific matrix Riemann-Hilbert problem is constructed for this nonlocal equation.Through solving this associated Riemann-Hilbert problem,the N-soliton solutions to this nonlocal equation can be obtained in the case of the jump matrix as an identity matrix.
基金supported by the Key Project of the NSFC(12131010)the NSFC(11771155,12271032)+1 种基金the NSF of Guangdong Province(2021A1515010249,2021A1515010303)supported by the NSFC(11971179,12371205)。
文摘This paper is concerned with the Navier-Stokes/Allen-Cahn system,which is used to model the dynamics of immiscible two-phase flows.We consider a 1D free boundary problem and assume that the viscosity coefficient depends on the density in the form ofη(ρ)=ρ^(α).The existence of unique global H^(2m)-solutions(m∈N)to the free boundary problem is proven for when 0<α<1/4.Furthermore,we obtain the global C^(∞)-solutions if the initial data is smooth.
基金financially supported by the National Key Research and Development Program of China(No.2021YFB3803101)the National Natural Science Foundation of China(Nos.52022011,51974028,and 52090041)+1 种基金the Xiaomi Young Scholars ProgramChina National Postdoctoral Program for Innovative Talents(No.BX20230042)。
文摘Solid solution-strengthened copper alloys have the advantages of a simple composition and manufacturing process,high mechanical and electrical comprehensive performances,and low cost;thus,they are widely used in high-speed rail contact wires,electronic component connectors,and other devices.Overcoming the contradiction between low alloying and high performance is an important challenge in the development of solid solution-strengthened copper alloys.Taking the typical solid solution-strengthened alloy Cu-4Zn-1Sn as the research object,we proposed using the element In to replace Zn and Sn to achieve low alloying in this work.Two new alloys,Cu-1.5Zn-1Sn-0.4In and Cu-1.5Zn-0.9Sn-0.6In,were designed and prepared.The total weight percentage content of alloying elements decreased by 43%and 41%,respectively,while the product of ultimate tensile strength(UTS)and electrical conductivity(EC)of the annealed state increased by 14%and 15%.After cold rolling with a 90%reduction,the UTS of the two new alloys reached 576 and 627MPa,respectively,the EC was 44.9%IACS and 42.0%IACS,and the product of UTS and EC(UTS×EC)was 97%and 99%higher than that of the annealed state alloy.The dislocations proliferated greatly in cold-rolled alloys,and the strengthening effects of dislocations reached 332 and 356 MPa,respectively,which is the main reason for the considerable improvement in mechanical properties.
基金Project supported by the National Natural Science Foundation of China (Grant Nos.12275172 and 11905124)。
文摘This paper studies the(2+1)-dimensional Hirota-Satsuma-Ito equation.Based on an associated Hirota bilinear form,lump-type solution,two types of interaction solutions,and breather wave solution of the(2+1)-dimensional Hirota-Satsuma-Ito equation are obtained,which are all related to the seed solution of the equation.It is interesting that the rogue wave is aroused by the interaction between one-lump soliton and a pair of resonance stripe solitons,and the fusion and fission phenomena are also found in the interaction between lump solitons and one-stripe soliton.Furthermore,the breather wave solution is also obtained by reducing the two-soliton solutions.The trajectory and period of the one-order breather wave are analyzed.The corresponding dynamical characteristics are demonstrated by the graphs.
基金supported by the National Natural Science Foundations of China(Grant Nos.12372073 and U20B2013)the Natural Science Basic Research Program of Shaanxi(Program No.2023-JC-QN-0030).
文摘A numerical approach is an effective means of solving boundary value problems(BVPs).This study focuses on physical problems with general partial differential equations(PDEs).It investigates the solution approach through the standard forms of the PDE module in COMSOL.Two typical mechanics problems are exemplified:The deflection of a thin plate,which can be addressed with the dedicated finite element module,and the stress of a pure bending beamthat cannot be tackled.The procedure for the two problems regarding the three standard forms required by the PDE module is detailed.The results were in good agreement with the literature,indicating that the PDE module provides a promising means to solve complex PDEs,especially for those a dedicated finite element module has yet to be developed.
