Periodic solutions of the Zakharov equation are investigated.By performing the limit operationλ_(2l-1)→λ_(1)on the eigenvalues of the Lax pair obtained from the n-fold Darboux transformation,an order-n breather-pos...Periodic solutions of the Zakharov equation are investigated.By performing the limit operationλ_(2l-1)→λ_(1)on the eigenvalues of the Lax pair obtained from the n-fold Darboux transformation,an order-n breather-positon solution is first obtained from a plane wave seed.It is then proven that an order-n lump solution can be further constructed by taking the limitλ_(1)→λ_(0)on the breather-positon solution,because the unique eigenvalueλ_(0)associated with the Lax pair eigenfunctionΨ(λ_(0))=0 corresponds to the limit of the infinite-periodic solutions.A convenient procedure of generating higher-order lump solutions of the Zakharov equation is also investigated based on the idea of the degeneration of double eigenvalues in multi-breather solutions.展开更多
The nonlinear Schrodinger equation is a classical integrable equation which contains plenty of significant properties and occurs in many physical areas.However,due to the difficulty of solving this equation,in particu...The nonlinear Schrodinger equation is a classical integrable equation which contains plenty of significant properties and occurs in many physical areas.However,due to the difficulty of solving this equation,in particular in high dimensions,lots of methods are proposed to effectively obtain different kinds of solutions,such as neural networks among others.Recently,a method where some underlying physical laws are embeded into a conventional neural network is proposed to uncover the equation’s dynamical behaviors from spatiotemporal data directly.Compared with traditional neural networks,this method can obtain remarkably accurate solution with extraordinarily less data.Meanwhile,this method also provides a better physical explanation and generalization.In this paper,based on the above method,we present an improved deep learning method to recover the soliton solutions,breather solution,and rogue wave solutions of the nonlinear Schrodinger equation.In particular,the dynamical behaviors and error analysis about the one-order and two-order rogue waves of nonlinear integrable equations are revealed by the deep neural network with physical constraints for the first time.Moreover,the effects of different numbers of initial points sampled,collocation points sampled,network layers,neurons per hidden layer on the one-order rogue wave dynamics of this equation have been considered with the help of the control variable way under the same initial and boundary conditions.Numerical experiments show that the dynamical behaviors of soliton solutions,breather solution,and rogue wave solutions of the integrable nonlinear Schrodinger equation can be well reconstructed by utilizing this physically-constrained deep learning method.展开更多
We derive an N-fold Darboux transformation for the nonlinear Schrdinger equation coupled to a multiple selfinduced transparency system, which is applicable to optical fiber communications in the erbium-doped medium.Th...We derive an N-fold Darboux transformation for the nonlinear Schrdinger equation coupled to a multiple selfinduced transparency system, which is applicable to optical fiber communications in the erbium-doped medium.The N-soliton, N-breather and N th-order rogue wave solutions in the compact determinant representations are derived using the Darboux transformation and limit technique. Dynamics of such solutions from the first-to second-order ones are shown.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
Solving nonlinear partial differential equations have attracted intensive attention in the past few decades. In this paper, the Darboux transformation method is used to derive several positon and hybrid solutions for ...Solving nonlinear partial differential equations have attracted intensive attention in the past few decades. In this paper, the Darboux transformation method is used to derive several positon and hybrid solutions for the(2+1)-dimensional complex modified Korteweg–de Vries equations. Based on the zero seed solution, the positon solution and the hybrid solutions of positon and soliton are constructed. The composition of positons is studied, showing that multi-positons of(2+1)-dimensional equations are decomposed into multi-solitons as well as the(1+1)-dimensions. Moreover, the interactions between positon and soliton are analyzed. In addition, the hybrid solutions of b-positon and breather are obtained using the plane wave seed solution, and their evolutions with time are discussed.展开更多
Based on the Hirota bilinear method,this study derived N-soliton solutions,breather solutions,lump solutions and interaction solutions for the(2+1)-dimensional extended Boiti-Leon-Manna-Pempinelli equation.The dynamic...Based on the Hirota bilinear method,this study derived N-soliton solutions,breather solutions,lump solutions and interaction solutions for the(2+1)-dimensional extended Boiti-Leon-Manna-Pempinelli equation.The dynamical characteristics of these solutions were displayed through graphical,particularly revealing fusion and ssion phenomena in the interaction of lump and the one-stripe soliton.展开更多
The behavior of a chemical tanker(CT)in extreme waves was discussed in detail,that is,in terms of rigid body heave and pitch motions,vertical bending moments(VBMs)amidships,green water,and slamming impacts through the...The behavior of a chemical tanker(CT)in extreme waves was discussed in detail,that is,in terms of rigid body heave and pitch motions,vertical bending moments(VBMs)amidships,green water,and slamming impacts through the analysis of the experimental data from model tests.Regular wave tests conducted for two wave steepness showed that the increase in wave steepness caused the increase in the asymmetry between hogging and sagging moments and the contribution of green water on deck to the decrease in vertical wave bending moments.Random uncertainty analysis of statistical values in irregular wave tests with various seeds revealed slight experimental uncertainties on motions and VBMs and slightly higher errors in slamming pressure peaks.With the increase in forward speed,experimental uncertainty on slamming pressures at the bow increased.Breather solutions of the nonlinear Schrödinger equation applied to generate tailored extreme waves of certain critical wavelengths showed a good performance in terms of ship response,and it was further verified for the CT.展开更多
The degradation and nonlinear interactions of a two-breather solution of the Mel’nikov equation are analyzed.By modulating the phase shift and limit method,we prove that in different regions near the non-singular bou...The degradation and nonlinear interactions of a two-breather solution of the Mel’nikov equation are analyzed.By modulating the phase shift and limit method,we prove that in different regions near the non-singular boundaries,there are four kinds of solutions with repulsive interaction or attractive interaction in addition to the two-breather solution.They are the interaction solution between soliton and breather,the two-soliton solution,and the two-breather solution with small amplitude,which all exhibit repulsive interactions;and the two-breather solution with small amplitude,which exhibits attractive interaction.Interestingly,a new breather acts as a messenger to transfer energy during the interaction between two breather solutions with small amplitude.展开更多
This paper presents a comprehensive experimental study on the effect of extreme waves on a LNG carrier.The LNG carrier model was equipped with a variety of sensors to measure motions,green water height on deck as well...This paper presents a comprehensive experimental study on the effect of extreme waves on a LNG carrier.The LNG carrier model was equipped with a variety of sensors to measure motions,green water height on deck as well as local and global loads.Experiments in transient wave packets provided the general performance in waves in terms of response amplitude operators and were accompanied by tests in regular waves with two different wave steepness.These tests allowed detailed insights into the nonlinear behavior of the vertical wave bending moment in steep waves showing that green water on deck can contribute to a decrease of vertical wave bending moment.Afterwards,systematic model tests in irregular waves were performed to provide the basis for statistical analysis.It is shown that the generalized extreme value distribution model is suitable for the estimation of the extreme peak values of motions and loads.Finally,model tests in tailored extreme wave sequences were conducted comparing the results with the statistical analysis.For this purpose,analytical breather solutions of the nonlinear Schrödinger equation were applied to generate tailored extreme waves of certain critical wave lengths in terms of ship response.Besides these design extreme waves,the LGN carrier was also investigated in the model scale reproduction of the real-world Draupner wave.By comparing the motions,vertical wave bending moment,green water column and slamming pressures it is concluded that the breather solutions are a powerful and efficient tool for the generation of design extreme waves of certain critical wave lengths for wave/structure investigations on different subjects.展开更多
基金sponsored by NUPTSF (Grant Nos.NY220161 and NY222169)the Foundation of Jiangsu Provincial Double-Innovation Doctor Program (Grant No.JSSCBS20210541)+1 种基金the Natural Science Foundation of the Higher Education Institutions of Jiangsu Province (Grant No.22KJB110004)the National Natural Science Foundation of China (Grant No.12171433)。
文摘Periodic solutions of the Zakharov equation are investigated.By performing the limit operationλ_(2l-1)→λ_(1)on the eigenvalues of the Lax pair obtained from the n-fold Darboux transformation,an order-n breather-positon solution is first obtained from a plane wave seed.It is then proven that an order-n lump solution can be further constructed by taking the limitλ_(1)→λ_(0)on the breather-positon solution,because the unique eigenvalueλ_(0)associated with the Lax pair eigenfunctionΨ(λ_(0))=0 corresponds to the limit of the infinite-periodic solutions.A convenient procedure of generating higher-order lump solutions of the Zakharov equation is also investigated based on the idea of the degeneration of double eigenvalues in multi-breather solutions.
基金supported by the National Natural Science Foundation of China (Grant No. 11675054)the Fund from Shanghai Collaborative Innovation Center of Trustworthy Software for Internet of Things (Grant No. ZF1213)the Project of Science and Technology Commission of Shanghai Municipality (Grant No. 18dz2271000)。
文摘The nonlinear Schrodinger equation is a classical integrable equation which contains plenty of significant properties and occurs in many physical areas.However,due to the difficulty of solving this equation,in particular in high dimensions,lots of methods are proposed to effectively obtain different kinds of solutions,such as neural networks among others.Recently,a method where some underlying physical laws are embeded into a conventional neural network is proposed to uncover the equation’s dynamical behaviors from spatiotemporal data directly.Compared with traditional neural networks,this method can obtain remarkably accurate solution with extraordinarily less data.Meanwhile,this method also provides a better physical explanation and generalization.In this paper,based on the above method,we present an improved deep learning method to recover the soliton solutions,breather solution,and rogue wave solutions of the nonlinear Schrodinger equation.In particular,the dynamical behaviors and error analysis about the one-order and two-order rogue waves of nonlinear integrable equations are revealed by the deep neural network with physical constraints for the first time.Moreover,the effects of different numbers of initial points sampled,collocation points sampled,network layers,neurons per hidden layer on the one-order rogue wave dynamics of this equation have been considered with the help of the control variable way under the same initial and boundary conditions.Numerical experiments show that the dynamical behaviors of soliton solutions,breather solution,and rogue wave solutions of the integrable nonlinear Schrodinger equation can be well reconstructed by utilizing this physically-constrained deep learning method.
基金Supported by the National Natural Science Foundation of China under Grant Nos 11705290 and 11305060the China Postdoctoral Science Foundation under Grant No 2016M602252
文摘We derive an N-fold Darboux transformation for the nonlinear Schrdinger equation coupled to a multiple selfinduced transparency system, which is applicable to optical fiber communications in the erbium-doped medium.The N-soliton, N-breather and N th-order rogue wave solutions in the compact determinant representations are derived using the Darboux transformation and limit technique. Dynamics of such solutions from the first-to second-order ones are shown.
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Project sponsored by NUPTSF(Grant Nos.NY220161and NY222169)the Foundation of Jiangsu Provincial Double-Innovation Doctor Program(Grant No.JSSCBS20210541)+1 种基金the Natural Science Foundation of the Higher Education Institutions of Jiangsu Province,China(Grant No.22KJB110004)the National Natural Science Foundation of China(Grant No.11871446)。
文摘Solving nonlinear partial differential equations have attracted intensive attention in the past few decades. In this paper, the Darboux transformation method is used to derive several positon and hybrid solutions for the(2+1)-dimensional complex modified Korteweg–de Vries equations. Based on the zero seed solution, the positon solution and the hybrid solutions of positon and soliton are constructed. The composition of positons is studied, showing that multi-positons of(2+1)-dimensional equations are decomposed into multi-solitons as well as the(1+1)-dimensions. Moreover, the interactions between positon and soliton are analyzed. In addition, the hybrid solutions of b-positon and breather are obtained using the plane wave seed solution, and their evolutions with time are discussed.
基金Supported by the National Natural Science Foundation of China(12275172)。
文摘Based on the Hirota bilinear method,this study derived N-soliton solutions,breather solutions,lump solutions and interaction solutions for the(2+1)-dimensional extended Boiti-Leon-Manna-Pempinelli equation.The dynamical characteristics of these solutions were displayed through graphical,particularly revealing fusion and ssion phenomena in the interaction of lump and the one-stripe soliton.
文摘The behavior of a chemical tanker(CT)in extreme waves was discussed in detail,that is,in terms of rigid body heave and pitch motions,vertical bending moments(VBMs)amidships,green water,and slamming impacts through the analysis of the experimental data from model tests.Regular wave tests conducted for two wave steepness showed that the increase in wave steepness caused the increase in the asymmetry between hogging and sagging moments and the contribution of green water on deck to the decrease in vertical wave bending moments.Random uncertainty analysis of statistical values in irregular wave tests with various seeds revealed slight experimental uncertainties on motions and VBMs and slightly higher errors in slamming pressure peaks.With the increase in forward speed,experimental uncertainty on slamming pressures at the bow increased.Breather solutions of the nonlinear Schrödinger equation applied to generate tailored extreme waves of certain critical wavelengths showed a good performance in terms of ship response,and it was further verified for the CT.
基金Project supported by the National Natural Science Foundation of China(Grant Nos.52171251 and U21062251)Program of Science and Technology Innovation of Dalian(Grant No.2022JJ12GX036).
文摘The degradation and nonlinear interactions of a two-breather solution of the Mel’nikov equation are analyzed.By modulating the phase shift and limit method,we prove that in different regions near the non-singular boundaries,there are four kinds of solutions with repulsive interaction or attractive interaction in addition to the two-breather solution.They are the interaction solution between soliton and breather,the two-soliton solution,and the two-breather solution with small amplitude,which all exhibit repulsive interactions;and the two-breather solution with small amplitude,which exhibits attractive interaction.Interestingly,a new breather acts as a messenger to transfer energy during the interaction between two breather solutions with small amplitude.
基金the experimental work performed during the project EXTREME SEASwhich was funded by the European Commissionunder the Grant agreement No. 234175
文摘This paper presents a comprehensive experimental study on the effect of extreme waves on a LNG carrier.The LNG carrier model was equipped with a variety of sensors to measure motions,green water height on deck as well as local and global loads.Experiments in transient wave packets provided the general performance in waves in terms of response amplitude operators and were accompanied by tests in regular waves with two different wave steepness.These tests allowed detailed insights into the nonlinear behavior of the vertical wave bending moment in steep waves showing that green water on deck can contribute to a decrease of vertical wave bending moment.Afterwards,systematic model tests in irregular waves were performed to provide the basis for statistical analysis.It is shown that the generalized extreme value distribution model is suitable for the estimation of the extreme peak values of motions and loads.Finally,model tests in tailored extreme wave sequences were conducted comparing the results with the statistical analysis.For this purpose,analytical breather solutions of the nonlinear Schrödinger equation were applied to generate tailored extreme waves of certain critical wave lengths in terms of ship response.Besides these design extreme waves,the LGN carrier was also investigated in the model scale reproduction of the real-world Draupner wave.By comparing the motions,vertical wave bending moment,green water column and slamming pressures it is concluded that the breather solutions are a powerful and efficient tool for the generation of design extreme waves of certain critical wave lengths for wave/structure investigations on different subjects.
