A new nonlinear dispersion relation is given in this paper, which can overcome the limitation of the intermediate minimum value in the dispersion relation proposed by Kirby and Dalrymple (1986). and which has a better...A new nonlinear dispersion relation is given in this paper, which can overcome the limitation of the intermediate minimum value in the dispersion relation proposed by Kirby and Dalrymple (1986). and which has a better approximation to Hedges' empirical relation than the modified relations by Hedges (1987). Kirby and Dalrymple (1987) for shallow waters. The new dispersion relation is simple in form, thus it can be used easily in practice. Meanwhile, a general explicit approximation to the new dispersion and other and other nonlinear dispersion relations is given. By use of the explicit approximation to the new dispersion relation along with the mild slope equation taking into account weakly nonlinenr effect, a mathematical model is obtained, and it is applied to laboratory data. The results show that the model developed with the new dispersion relation predicts wave transformation over complicated topography quite well.展开更多
A nonlinear dispersion relation is presented to model the nonlinear dispersion of waves over the whole range of possible water depths. It reduces the phase speed over prediction of both Hedges′ modified relation and...A nonlinear dispersion relation is presented to model the nonlinear dispersion of waves over the whole range of possible water depths. It reduces the phase speed over prediction of both Hedges′ modified relation and Kirby and Dalrymple′s modified relation in the region of 1< kh <1 5 for small wave steepness and maintains the monotonicity in phase speed variation for large wave steepness. And it has a simple form. By use of the new nonlinear dispersion relation along with the mild slope equation taking into account weak nonlinearity, a mathematical model of wave transformation is developed and applied to laboratory data. The results show that the model with the new dispersion relation can predict wave transformation over complicated bathymetry satisfactorily.展开更多
The nonlinear dispersion relations and modified relations proposed by Kirby and Hedges have the limitation of intermediate minimum value. To overcome the shortcoming, a new nonlinear dispersion relation is proposed. B...The nonlinear dispersion relations and modified relations proposed by Kirby and Hedges have the limitation of intermediate minimum value. To overcome the shortcoming, a new nonlinear dispersion relation is proposed. Based on the summarization and comparison of existing nonlinear dispersion relations, it can be found that the new nonlinear dispersion relation not only keeps the advantages of other nonlinear dispersion relations, but also significantly reduces the relative errors of the nonlinear dispersion relations for a range of the relative water depth of 1<kh<1.5 and has sufficient accuracy for practical purposes.展开更多
Based on the hyperbolic mild-slope equations derived by Copeland (1985), a numerical model is established in unstag- gered grids. A composite 4 th-order Adam-Bashforth-Moulton (ABM) scheme is used to solve the model i...Based on the hyperbolic mild-slope equations derived by Copeland (1985), a numerical model is established in unstag- gered grids. A composite 4 th-order Adam-Bashforth-Moulton (ABM) scheme is used to solve the model in the time domain. Terms involving the first order spatial derivatives are differenced to O ( Δx )4accuracy utilizing a five-point formula. The nonlinear dispersion relationship proposed by Kirby and Dalrymple (1986) is used to consider the nonlinear effect. A numerical test is performed upon wave propagating over a typical shoal. The agreement between the numerical and the experimental results validates the present model. Biodistribution and applications are also summarized.展开更多
The present paper aims to investigate the chirped optical soliton solutions of the nonlinear Schrödinger equation with nonlinear chromatic dispersion and quadratic-cubic law of refractive index.The exquisite bala...The present paper aims to investigate the chirped optical soliton solutions of the nonlinear Schrödinger equation with nonlinear chromatic dispersion and quadratic-cubic law of refractive index.The exquisite balance between the chromatic dispersion and the nonlinearity associated with the refractive index of a fiber gives rise to optical solitons,which can travel down the fiber for intercontinental distances.The effective technique,namely,the new extended auxiliary equation method is implemented as a solution method.Different types of chirped soliton solutions including dark,bright,singular and periodic soliton solutions are extracted from the Jacobi elliptic function solutions when the modulus of the Jacobi elliptic function approaches to one or zero.These obtained chirped optical soliton solutions might play an important role in optical communication links and optical signal processing systems.The stability of the system is examined in the framework of modulational instability analysis.展开更多
In the present paper, by introducing the effective wave elevation, we transform the extended elliptic mild-slope equation with bottom friction, wave breaking and steep or rapidly varying bottom topography to the simpl...In the present paper, by introducing the effective wave elevation, we transform the extended elliptic mild-slope equation with bottom friction, wave breaking and steep or rapidly varying bottom topography to the simplest time-dependent hyperbolic equation. Based on this equation and the empirical nonlinear amplitude dispersion relation proposed by Li et al. （2003）, the numerical scheme is established. Error analysis by Taylor expansion method shows that the numerical stability of the present model succeeds the merits in Song et al. （2007）＇s model because of the introduced dissipation terms. For the purpose of verifying its performance on wave nonlinearity, rapidly varying topography and wave breaking, the present model is applied to study： （1） wave refraction and diffraction over a submerged elliptic shoal on a slope （Berkhoff et al., 1982）; （2） Bragg reflection of monochromatic waves from the sinusoidal ripples （Davies and Heathershaw, 1985）; （3） wave transformation near a shore attached breakwater （Watanabe and Maruyama, 1986）. Comparisons of the numerical solutions with the experimental or theoretical ones or with those of other models （REF/DIF model and FUNWAVE model） show good results, which indicate that the present model is capable of giving favorably predictions of wave refraction, diffraction, reflection, shoaling, bottom friction, breaking energy dissipation and weak nonlinearity in the near shore zone.展开更多
Nonlinear effect is of importance to waves propagating from deep water to shallow water. The non-linearity of waves is widely discussed due to its high precision in application. But there are still some problems in de...Nonlinear effect is of importance to waves propagating from deep water to shallow water. The non-linearity of waves is widely discussed due to its high precision in application. But there are still some problems in dealing with the nonlinear waves in practice. In this paper, a modified form of mild-slope equation with weakly nonlinear effect is derived by use of the nonlinear dispersion relation and the steady mild-slope equation containing energy dissipation. The modified form of mild-slope equation is convenient to solve nonlinear effect of waves. The model is tested against the laboratory measurement for the case of a submerged elliptical shoal on a slope beach given by Berkhoff et al. The present numerical results are also compared with those obtained through linear wave theory. Better agreement is obtained as the modified mild-slope equation is employed. And the modified mild-slope equation can reasonably simulate the weakly nonlinear effect of wave propagation from deep water to coast.展开更多
In this paper,the bifurcation of solitary,kink,anti-kink,and periodic waves for(2+1)-dimension nonlinear dispersive long wave equation is studied by using the bifurcation theory of planar dynamical systems.Bifurcation...In this paper,the bifurcation of solitary,kink,anti-kink,and periodic waves for(2+1)-dimension nonlinear dispersive long wave equation is studied by using the bifurcation theory of planar dynamical systems.Bifurcation parameter sets are shown,and under various parameter conditions,all exact explicit formulas of solitary travelling wave solutions and kink travelling wave solutions and periodic travelling wave solutions are listed.展开更多
In this paper,the rogue waves of the higher-order dispersive nonlinear Schrdinger(HDNLS) equation are investigated,which describes the propagation of ultrashort optical pulse in optical fibers.The rogue wave solutions...In this paper,the rogue waves of the higher-order dispersive nonlinear Schrdinger(HDNLS) equation are investigated,which describes the propagation of ultrashort optical pulse in optical fibers.The rogue wave solutions of HDNLS equation are constructed by using the modified Darboux transformation method.The explicit first and secondorder rogue wave solutions are presented under the plane wave seeding solution background.The nonlinear dynamics and properties of rogue waves are discussed by analyzing the obtained rational solutions.The influence of little perturbation on the rogue waves is discussed with the help of graphical simulation.展开更多
Abstract By making use of the generalized sine-Gordon equation expansion method, we lind cnoidal periodic wave solutions and fundamental bright and dark optical solitary wave solutions for the fourth-order dispersive ...Abstract By making use of the generalized sine-Gordon equation expansion method, we lind cnoidal periodic wave solutions and fundamental bright and dark optical solitary wave solutions for the fourth-order dispersive and the quintic nonlinear Schroedinger equation with self-steepening, and self-frequency shift. Moreover, we discuss the formation conditions of the bright and dark solitary waves.展开更多
For the (2 + 1)-dimensional nonlinear dispersive Boussinesq equation, by using the bifurcation theory of planar dynamical systems to study its corresponding traveling wave system, the bifurcations and phase portraits ...For the (2 + 1)-dimensional nonlinear dispersive Boussinesq equation, by using the bifurcation theory of planar dynamical systems to study its corresponding traveling wave system, the bifurcations and phase portraits of the regular system are obtained. Under different parametric conditions, various sufficient conditions to guarantee the existence of analytical and non-analytical solutions of the singular system are given by using singular traveling wave theory. For certain special cases, some explicit and exact parametric representations of traveling wave solutions are derived such as analytical periodic waves and non-analytical periodic cusp waves. Further, two-dimensional wave plots of analytical periodic solutions and non-analytical periodic cusp wave solutions are drawn to visualize the dynamics of the equation.展开更多
The original hyperbolic mild-slope equation can effectively take into account the combined effects of wave shoaling, refraction, diffraction and reflection, but does not consider the nonlinear effect of waves, and the...The original hyperbolic mild-slope equation can effectively take into account the combined effects of wave shoaling, refraction, diffraction and reflection, but does not consider the nonlinear effect of waves, and the existing numerical schemes for it show some deficiencies. Based on the original hyperbolic mild-slope equation, a nonlinear dispersion relation is introduced in present paper to effectively take the nonlinear effect of waves into account and a new numerical scheme is proposed. The weakly nonlinear dispersion relation and the improved numerical scheme are applied to the simulation of wave transformation over an elliptic shoal. Numerical tests show that the improvement of the numerical scheme makes efficient the solution to the hyperbolic mild-slope equation, A comparison of numerical results with experimental data indicates that the results obtained by use of the new scheme are satisfactory.展开更多
By considering higher-order effects, the properties of self-similar parabolic pulses propagating in the microstructured fibre amplifier with a normal group-velocity dispersion have been investigated. The numerical res...By considering higher-order effects, the properties of self-similar parabolic pulses propagating in the microstructured fibre amplifier with a normal group-velocity dispersion have been investigated. The numerical results indicate that the higher-order effects can badly distort self-similar parabolic pulse shape and optical spectrum, and at the same time the peak shift and oscillation appear, while the pulse still reveals highly linear chirp but grows into asymmetry. The influence of different higher-order effects on self-similar parabolic pulse propagation has been analysed. It shows that the self-steepening plays a more important role. We can manipulate the geometrical parameters of the microstructured fibre amplifier to gain a suitable dispersion and nonlinearity coefficient which will keep high-quality self-similar parabolic pulse propagation. These results are significant for the further study of self-similar parabolic pulse propagation.展开更多
A mode-locked thulium-doped fiber laser(TDFL) based on nonlinear polarization rotation(NPR) with different net anomalous dispersion is demonstrated. When the cavity dispersion is-1.425 ps^2, the noise-like(NL) pulse w...A mode-locked thulium-doped fiber laser(TDFL) based on nonlinear polarization rotation(NPR) with different net anomalous dispersion is demonstrated. When the cavity dispersion is-1.425 ps^2, the noise-like(NL) pulse with coherence spike width of 406 fs and pulse energy of 12.342 nJ is generated at a center wavelength of 2003.2 nm with 3 dB spectral bandwidth of 23.20 nm. In the experimental period of 400 min, the 3 dB spectral bandwidth variation, the output power fluctuation, and the central wavelength shift are less than 0.06 nm, 0.04 d B, and0.4 nm, respectively, indicating that the NPR-based TDFL operating in the NL regime holds good long-term stability.展开更多
By the use of an auxiliary equation, we find bright and dark optical soliton and other soliton solutions for the higher-order nonlinear Schrodinger equation (NLSE) with fourth-order dispersion (FOD), cubic-quintic ter...By the use of an auxiliary equation, we find bright and dark optical soliton and other soliton solutions for the higher-order nonlinear Schrodinger equation (NLSE) with fourth-order dispersion (FOD), cubic-quintic terms, self-steepening, and nonlinear dispersive terms. Moreover, we give the formation condition of the bright and dark solitons for this higher-order NLSE.展开更多
Semiparametric reproductive dispersion nonlinear model (SRDNM) is an extension of nonlinear reproductive dispersion models and semiparametric nonlinear regression models, and includes semiparametric nonlinear model an...Semiparametric reproductive dispersion nonlinear model (SRDNM) is an extension of nonlinear reproductive dispersion models and semiparametric nonlinear regression models, and includes semiparametric nonlinear model and semiparametric generalized linear model as its special cases. Based on the local kernel estimate of nonparametric component, profile-kernel and backfitting estimators of parameters of interest are proposed in SRDNM, and theoretical comparison of both estimators is also investigated in this paper. Under some regularity conditions, strong consistency and asymptotic normality of two estimators are proved. It is shown that the backfitting method produces a larger asymptotic variance than that for the profile-kernel method. A simulation study and a real example are used to illustrate the proposed methodologies.展开更多
The interaction between three optical solitons is a complex and valuable research direction,which is of practical application for promoting the development of optical communication and all-optical information processi...The interaction between three optical solitons is a complex and valuable research direction,which is of practical application for promoting the development of optical communication and all-optical information processing technology.In this paper,we start from the study of the variable-coefficient coupled higher-order nonlinear Schodinger equation(VCHNLSE),and obtain an analytical three-soliton solution of this equation.Based on the obtained solution,the interaction of the three optical solitons is explored when they are incident from different initial velocities and phases.When the higher-order dispersion and nonlinear functions are sinusoidal,hyperbolic secant,and hyperbolic tangent functions,the transmission properties of three optical solitons before and after interactions are discussed.Besides,this paper achieves effective regulation of amplitude and velocity of optical solitons as well as of the local state of interaction process,and interaction-free transmission of the three optical solitons is obtained with a small spacing.The relevant conclusions of the paper are of great significance in promoting the development of high-speed and large-capacity optical communication,optical signal processing,and optical computing.展开更多
We propose a novel algorithm,based on physics-informed neural networks(PINNs)to efficiently approximate solutions of nonlinear dispersive PDEs such as the KdV-Kawahara,Camassa-Holm and Benjamin-Ono equations.The stabi...We propose a novel algorithm,based on physics-informed neural networks(PINNs)to efficiently approximate solutions of nonlinear dispersive PDEs such as the KdV-Kawahara,Camassa-Holm and Benjamin-Ono equations.The stability of solutions of these dispersive PDEs is leveraged to prove rigorous bounds on the resulting error.We present several numerical experiments to demonstrate that PINNs can approximate solutions of these dispersive PDEs very accurately.展开更多
This paper presents the development of a theoretical model of fully nonlinear and weakly dispersive(FNWD)waves and numerical techniques for simulating the propagation,interaction,and transformation of solitary waves.U...This paper presents the development of a theoretical model of fully nonlinear and weakly dispersive(FNWD)waves and numerical techniques for simulating the propagation,interaction,and transformation of solitary waves.Using the standard expansion method and without the limit of small nonlinear parameter defined as the ratio of the wave height versus water depth,a set of model equations describing the FNWD waves in a domain of moderately varying bottom topography are formulated.Exact solitary wave solutions satisfying the FNWD equations are also derived.Numerically,a time-accurate and stabilized finite-element code to solve the governing equations is developed for wave simulations.The solitary wave solutions of FNWD,weakly nonlinear and weakly dispersive(WNWD),and Laplace equations based models in terms of wave profile and phase speed are compared to examine their related features and differences.Investigations on the overtaking collision of two unidirectional solitary waves of different amplitudes,i.e.,ax and a2 where a1>a2,are carried out using both the FNWD and WNWD water wave models.Selected cases by running the FNWD and WNWD models are performed to identify the critical values of a1/a2 for forming a flattened merging wave peak,which is the condition used to determine if the stronger wave is to pass through the weaker one or both waves are to remain separated during the encountering process.It is interesting to note the critical values of a1/a2 obtained from the FNWD and WNWD models are found to be different and greater than the value of 3 proposed by Wu through the theoretical analysis of the Korteweg-de Vries(KdV)equations.Finally,the phenomena of wave splitting and nonlinear focusing of a solitary wave propagating over a three-dimensional semicircular shoal are simulated.The results obtained from both the FNWD and WNWD models showing the fission process of separating a main solitary wave into multiple waves of decreasing amplitudes are presented,compared,and discussed.展开更多
In this paper,an efficient hybrid numerical scheme which is based on a joint venture of the q-homotopy analysis method and Sumudu transform is applied to investigate the time-fractional modified Degasperis-Procesi(DP)...In this paper,an efficient hybrid numerical scheme which is based on a joint venture of the q-homotopy analysis method and Sumudu transform is applied to investigate the time-fractional modified Degasperis-Procesi(DP)equation.The present study considers the Caputo fractional derivative.The fractional order modified DP model is very important and plays a great role in study of ocean engineering and science.The proposed scheme provides a beautiful opportunity for proper selection of the auxiliary parameter h and the asymptotic parameterρ(≥1)to handle mainly the differential equations of nonlinear nature.The offered scheme produces the solution in the shape of a convergent series in a large admissible domain which is helpful to regulate the region of convergence of a series solution.The proposed work computes the approximate analytical solution of the fractional modified DP equation systematically and also presents graphically the variation of the obtained solution for diverse values of the fractional parameterβ.展开更多
文摘A new nonlinear dispersion relation is given in this paper, which can overcome the limitation of the intermediate minimum value in the dispersion relation proposed by Kirby and Dalrymple (1986). and which has a better approximation to Hedges' empirical relation than the modified relations by Hedges (1987). Kirby and Dalrymple (1987) for shallow waters. The new dispersion relation is simple in form, thus it can be used easily in practice. Meanwhile, a general explicit approximation to the new dispersion and other and other nonlinear dispersion relations is given. By use of the explicit approximation to the new dispersion relation along with the mild slope equation taking into account weakly nonlinenr effect, a mathematical model is obtained, and it is applied to laboratory data. The results show that the model developed with the new dispersion relation predicts wave transformation over complicated topography quite well.
文摘A nonlinear dispersion relation is presented to model the nonlinear dispersion of waves over the whole range of possible water depths. It reduces the phase speed over prediction of both Hedges′ modified relation and Kirby and Dalrymple′s modified relation in the region of 1< kh <1 5 for small wave steepness and maintains the monotonicity in phase speed variation for large wave steepness. And it has a simple form. By use of the new nonlinear dispersion relation along with the mild slope equation taking into account weak nonlinearity, a mathematical model of wave transformation is developed and applied to laboratory data. The results show that the model with the new dispersion relation can predict wave transformation over complicated bathymetry satisfactorily.
基金This work was financially supported by the Key Project of National Natural Science Foundation of China (Grant No.50339010) and the Key Project of Chinese Ministry of Education (Grant No.03095)
文摘The nonlinear dispersion relations and modified relations proposed by Kirby and Hedges have the limitation of intermediate minimum value. To overcome the shortcoming, a new nonlinear dispersion relation is proposed. Based on the summarization and comparison of existing nonlinear dispersion relations, it can be found that the new nonlinear dispersion relation not only keeps the advantages of other nonlinear dispersion relations, but also significantly reduces the relative errors of the nonlinear dispersion relations for a range of the relative water depth of 1<kh<1.5 and has sufficient accuracy for practical purposes.
基金This work is financially supported by the National Natural Science Foundation of China （50409015）.
文摘Based on the hyperbolic mild-slope equations derived by Copeland (1985), a numerical model is established in unstag- gered grids. A composite 4 th-order Adam-Bashforth-Moulton (ABM) scheme is used to solve the model in the time domain. Terms involving the first order spatial derivatives are differenced to O ( Δx )4accuracy utilizing a five-point formula. The nonlinear dispersion relationship proposed by Kirby and Dalrymple (1986) is used to consider the nonlinear effect. A numerical test is performed upon wave propagating over a typical shoal. The agreement between the numerical and the experimental results validates the present model. Biodistribution and applications are also summarized.
文摘The present paper aims to investigate the chirped optical soliton solutions of the nonlinear Schrödinger equation with nonlinear chromatic dispersion and quadratic-cubic law of refractive index.The exquisite balance between the chromatic dispersion and the nonlinearity associated with the refractive index of a fiber gives rise to optical solitons,which can travel down the fiber for intercontinental distances.The effective technique,namely,the new extended auxiliary equation method is implemented as a solution method.Different types of chirped soliton solutions including dark,bright,singular and periodic soliton solutions are extracted from the Jacobi elliptic function solutions when the modulus of the Jacobi elliptic function approaches to one or zero.These obtained chirped optical soliton solutions might play an important role in optical communication links and optical signal processing systems.The stability of the system is examined in the framework of modulational instability analysis.
基金Open Fund of Key Laboratory of Coastal Disasters and Defence (Ministry of Education)National Natural Science Foundation of China under contract No. 50779015
文摘In the present paper, by introducing the effective wave elevation, we transform the extended elliptic mild-slope equation with bottom friction, wave breaking and steep or rapidly varying bottom topography to the simplest time-dependent hyperbolic equation. Based on this equation and the empirical nonlinear amplitude dispersion relation proposed by Li et al. （2003）, the numerical scheme is established. Error analysis by Taylor expansion method shows that the numerical stability of the present model succeeds the merits in Song et al. （2007）＇s model because of the introduced dissipation terms. For the purpose of verifying its performance on wave nonlinearity, rapidly varying topography and wave breaking, the present model is applied to study： （1） wave refraction and diffraction over a submerged elliptic shoal on a slope （Berkhoff et al., 1982）; （2） Bragg reflection of monochromatic waves from the sinusoidal ripples （Davies and Heathershaw, 1985）; （3） wave transformation near a shore attached breakwater （Watanabe and Maruyama, 1986）. Comparisons of the numerical solutions with the experimental or theoretical ones or with those of other models （REF/DIF model and FUNWAVE model） show good results, which indicate that the present model is capable of giving favorably predictions of wave refraction, diffraction, reflection, shoaling, bottom friction, breaking energy dissipation and weak nonlinearity in the near shore zone.
文摘Nonlinear effect is of importance to waves propagating from deep water to shallow water. The non-linearity of waves is widely discussed due to its high precision in application. But there are still some problems in dealing with the nonlinear waves in practice. In this paper, a modified form of mild-slope equation with weakly nonlinear effect is derived by use of the nonlinear dispersion relation and the steady mild-slope equation containing energy dissipation. The modified form of mild-slope equation is convenient to solve nonlinear effect of waves. The model is tested against the laboratory measurement for the case of a submerged elliptical shoal on a slope beach given by Berkhoff et al. The present numerical results are also compared with those obtained through linear wave theory. Better agreement is obtained as the modified mild-slope equation is employed. And the modified mild-slope equation can reasonably simulate the weakly nonlinear effect of wave propagation from deep water to coast.
基金Supported by the National Natural Science Foundation of China(10871206)Program for Excellent Talents in Guangxi Higher Education Institutions
文摘In this paper,the bifurcation of solitary,kink,anti-kink,and periodic waves for(2+1)-dimension nonlinear dispersive long wave equation is studied by using the bifurcation theory of planar dynamical systems.Bifurcation parameter sets are shown,and under various parameter conditions,all exact explicit formulas of solitary travelling wave solutions and kink travelling wave solutions and periodic travelling wave solutions are listed.
基金Supported by the National Natural Science Foundation of China under Grant No.11071164Innovation Program of Shanghai Municipal Education Commission under Grant Nos.12YZ105 and 13ZZ118+1 种基金the Foundation of University Young Teachers Training Program of Shanghai Municipal Education Commission under Grant No.slg11029the National Natural Science Foundation of China under Grant No.11171220
文摘In this paper,the rogue waves of the higher-order dispersive nonlinear Schrdinger(HDNLS) equation are investigated,which describes the propagation of ultrashort optical pulse in optical fibers.The rogue wave solutions of HDNLS equation are constructed by using the modified Darboux transformation method.The explicit first and secondorder rogue wave solutions are presented under the plane wave seeding solution background.The nonlinear dynamics and properties of rogue waves are discussed by analyzing the obtained rational solutions.The influence of little perturbation on the rogue waves is discussed with the help of graphical simulation.
基金The project supported by National Natural Science Foundation of Zhejiang Province of China under Grant No. Y605312
文摘Abstract By making use of the generalized sine-Gordon equation expansion method, we lind cnoidal periodic wave solutions and fundamental bright and dark optical solitary wave solutions for the fourth-order dispersive and the quintic nonlinear Schroedinger equation with self-steepening, and self-frequency shift. Moreover, we discuss the formation conditions of the bright and dark solitary waves.
文摘For the (2 + 1)-dimensional nonlinear dispersive Boussinesq equation, by using the bifurcation theory of planar dynamical systems to study its corresponding traveling wave system, the bifurcations and phase portraits of the regular system are obtained. Under different parametric conditions, various sufficient conditions to guarantee the existence of analytical and non-analytical solutions of the singular system are given by using singular traveling wave theory. For certain special cases, some explicit and exact parametric representations of traveling wave solutions are derived such as analytical periodic waves and non-analytical periodic cusp waves. Further, two-dimensional wave plots of analytical periodic solutions and non-analytical periodic cusp wave solutions are drawn to visualize the dynamics of the equation.
基金This subject was financially supported by the National Natural Science Foundation of China(Grant No. 59839330 and No.59976047) by the Visiting Scholal Foundation of State Key Hydraulic Lab.of High Speed Flows of Dalian University of Technology.
文摘The original hyperbolic mild-slope equation can effectively take into account the combined effects of wave shoaling, refraction, diffraction and reflection, but does not consider the nonlinear effect of waves, and the existing numerical schemes for it show some deficiencies. Based on the original hyperbolic mild-slope equation, a nonlinear dispersion relation is introduced in present paper to effectively take the nonlinear effect of waves into account and a new numerical scheme is proposed. The weakly nonlinear dispersion relation and the improved numerical scheme are applied to the simulation of wave transformation over an elliptic shoal. Numerical tests show that the improvement of the numerical scheme makes efficient the solution to the hyperbolic mild-slope equation, A comparison of numerical results with experimental data indicates that the results obtained by use of the new scheme are satisfactory.
基金Project supported by the National Science Foundation of Guangdong Province,China(Grant No04010397)
文摘By considering higher-order effects, the properties of self-similar parabolic pulses propagating in the microstructured fibre amplifier with a normal group-velocity dispersion have been investigated. The numerical results indicate that the higher-order effects can badly distort self-similar parabolic pulse shape and optical spectrum, and at the same time the peak shift and oscillation appear, while the pulse still reveals highly linear chirp but grows into asymmetry. The influence of different higher-order effects on self-similar parabolic pulse propagation has been analysed. It shows that the self-steepening plays a more important role. We can manipulate the geometrical parameters of the microstructured fibre amplifier to gain a suitable dispersion and nonlinearity coefficient which will keep high-quality self-similar parabolic pulse propagation. These results are significant for the further study of self-similar parabolic pulse propagation.
基金Fundamental Research Funds for the Central Universities(2016YJS034)
文摘A mode-locked thulium-doped fiber laser(TDFL) based on nonlinear polarization rotation(NPR) with different net anomalous dispersion is demonstrated. When the cavity dispersion is-1.425 ps^2, the noise-like(NL) pulse with coherence spike width of 406 fs and pulse energy of 12.342 nJ is generated at a center wavelength of 2003.2 nm with 3 dB spectral bandwidth of 23.20 nm. In the experimental period of 400 min, the 3 dB spectral bandwidth variation, the output power fluctuation, and the central wavelength shift are less than 0.06 nm, 0.04 d B, and0.4 nm, respectively, indicating that the NPR-based TDFL operating in the NL regime holds good long-term stability.
文摘By the use of an auxiliary equation, we find bright and dark optical soliton and other soliton solutions for the higher-order nonlinear Schrodinger equation (NLSE) with fourth-order dispersion (FOD), cubic-quintic terms, self-steepening, and nonlinear dispersive terms. Moreover, we give the formation condition of the bright and dark solitons for this higher-order NLSE.
基金supported by National Natural Science Foundation of China (Grant Nos. 10561008, 10761011)Natural Science Foundation of Department of Education of Zhejiang Province (Grant No. Y200805073)+1 种基金PhD Special Scientific Research Foundation of Chinese University (Grant No. 20060673002)Program for New Century Excellent Talents in University (Grant No. NCET-07-0737)
文摘Semiparametric reproductive dispersion nonlinear model (SRDNM) is an extension of nonlinear reproductive dispersion models and semiparametric nonlinear regression models, and includes semiparametric nonlinear model and semiparametric generalized linear model as its special cases. Based on the local kernel estimate of nonparametric component, profile-kernel and backfitting estimators of parameters of interest are proposed in SRDNM, and theoretical comparison of both estimators is also investigated in this paper. Under some regularity conditions, strong consistency and asymptotic normality of two estimators are proved. It is shown that the backfitting method produces a larger asymptotic variance than that for the profile-kernel method. A simulation study and a real example are used to illustrate the proposed methodologies.
基金supported by the Scientific Research Foundation of Weifang University of Science and Technology(Grant Nos.KJRC2022002 and KJRC2023035).
文摘The interaction between three optical solitons is a complex and valuable research direction,which is of practical application for promoting the development of optical communication and all-optical information processing technology.In this paper,we start from the study of the variable-coefficient coupled higher-order nonlinear Schodinger equation(VCHNLSE),and obtain an analytical three-soliton solution of this equation.Based on the obtained solution,the interaction of the three optical solitons is explored when they are incident from different initial velocities and phases.When the higher-order dispersion and nonlinear functions are sinusoidal,hyperbolic secant,and hyperbolic tangent functions,the transmission properties of three optical solitons before and after interactions are discussed.Besides,this paper achieves effective regulation of amplitude and velocity of optical solitons as well as of the local state of interaction process,and interaction-free transmission of the three optical solitons is obtained with a small spacing.The relevant conclusions of the paper are of great significance in promoting the development of high-speed and large-capacity optical communication,optical signal processing,and optical computing.
文摘We propose a novel algorithm,based on physics-informed neural networks(PINNs)to efficiently approximate solutions of nonlinear dispersive PDEs such as the KdV-Kawahara,Camassa-Holm and Benjamin-Ono equations.The stability of solutions of these dispersive PDEs is leveraged to prove rigorous bounds on the resulting error.We present several numerical experiments to demonstrate that PINNs can approximate solutions of these dispersive PDEs very accurately.
文摘This paper presents the development of a theoretical model of fully nonlinear and weakly dispersive(FNWD)waves and numerical techniques for simulating the propagation,interaction,and transformation of solitary waves.Using the standard expansion method and without the limit of small nonlinear parameter defined as the ratio of the wave height versus water depth,a set of model equations describing the FNWD waves in a domain of moderately varying bottom topography are formulated.Exact solitary wave solutions satisfying the FNWD equations are also derived.Numerically,a time-accurate and stabilized finite-element code to solve the governing equations is developed for wave simulations.The solitary wave solutions of FNWD,weakly nonlinear and weakly dispersive(WNWD),and Laplace equations based models in terms of wave profile and phase speed are compared to examine their related features and differences.Investigations on the overtaking collision of two unidirectional solitary waves of different amplitudes,i.e.,ax and a2 where a1>a2,are carried out using both the FNWD and WNWD water wave models.Selected cases by running the FNWD and WNWD models are performed to identify the critical values of a1/a2 for forming a flattened merging wave peak,which is the condition used to determine if the stronger wave is to pass through the weaker one or both waves are to remain separated during the encountering process.It is interesting to note the critical values of a1/a2 obtained from the FNWD and WNWD models are found to be different and greater than the value of 3 proposed by Wu through the theoretical analysis of the Korteweg-de Vries(KdV)equations.Finally,the phenomena of wave splitting and nonlinear focusing of a solitary wave propagating over a three-dimensional semicircular shoal are simulated.The results obtained from both the FNWD and WNWD models showing the fission process of separating a main solitary wave into multiple waves of decreasing amplitudes are presented,compared,and discussed.
文摘In this paper,an efficient hybrid numerical scheme which is based on a joint venture of the q-homotopy analysis method and Sumudu transform is applied to investigate the time-fractional modified Degasperis-Procesi(DP)equation.The present study considers the Caputo fractional derivative.The fractional order modified DP model is very important and plays a great role in study of ocean engineering and science.The proposed scheme provides a beautiful opportunity for proper selection of the auxiliary parameter h and the asymptotic parameterρ(≥1)to handle mainly the differential equations of nonlinear nature.The offered scheme produces the solution in the shape of a convergent series in a large admissible domain which is helpful to regulate the region of convergence of a series solution.The proposed work computes the approximate analytical solution of the fractional modified DP equation systematically and also presents graphically the variation of the obtained solution for diverse values of the fractional parameterβ.
