A nuclear explosion in the rock mass medium can produce strong shock waves,seismic shocks,and other destructive effects,which can cause extreme damage to the underground protection infrastructures.With the increase in...A nuclear explosion in the rock mass medium can produce strong shock waves,seismic shocks,and other destructive effects,which can cause extreme damage to the underground protection infrastructures.With the increase in nuclear explosion power,underground protection engineering enabled by explosion-proof impact theory and technology ushered in a new challenge.This paper proposes to simulate nuclear explosion tests with on-site chemical explosion tests in the form of multi-hole explosions.First,the mechanism of using multi-hole simultaneous blasting to simulate a nuclear explosion to generate approximate plane waves was analyzed.The plane pressure curve at the vault of the underground protective tunnel under the action of the multi-hole simultaneous blasting was then obtained using the impact test in the rock mass at the site.According to the peak pressure at the vault plane,it was divided into three regions:the stress superposition region,the superposition region after surface reflection,and the approximate plane stress wave zone.A numerical simulation approach was developed using PFC and FLAC to study the peak particle velocity in the surrounding rock of the underground protective cave under the action of multi-hole blasting.The time-history curves of pressure and peak pressure partition obtained by the on-site multi-hole simultaneous blasting test and numerical simulation were compared and analyzed,to verify the correctness and rationality of the formation of an approximate plane wave in the simulated nuclear explosion.This comparison and analysis also provided a theoretical foundation and some research ideas for the ensuing study on the impact of a nuclear explosion.展开更多
We investigate the rotating wave approximation applied in the high-spin quantum system driven by a linearly polarized alternating magnetic field in the presence of quadrupole interactions.The conventional way to apply...We investigate the rotating wave approximation applied in the high-spin quantum system driven by a linearly polarized alternating magnetic field in the presence of quadrupole interactions.The conventional way to apply the rotating wave approximation in a driven high-spin system is to assume the dynamics being restricted in the reduced Hilbert space.However,when the driving strength is relatively strong or the driving is off resonant,the leakage from the target resonance subspace cannot be neglected for a multi-level quantum system.We propose the correct formalism to apply the rotating wave approximation in the full Hilbert space by taking this leakage into account.By estimating the operator fidelity of the time propagator,our formalism applied in the full Hilbert space unambiguously manifests great advantages over the conventional method applied in the reduced Hilbert space.展开更多
We study the steady-state entanglement and heat current of two coupled qubits, in which two qubits are connected with two independent heat baths(IHBs) or two common heat baths(CHBs). We construct the master equation i...We study the steady-state entanglement and heat current of two coupled qubits, in which two qubits are connected with two independent heat baths(IHBs) or two common heat baths(CHBs). We construct the master equation in the eigenstate representation of two coupled qubits to describe the dynamics of the total system and derive the solutions in the steadystate with stronger coupling regime between two qubits than qubit–baths. We do not make the rotating wave approximation(RWA) for the qubit–qubit interaction, and so we are able to investigate the behaviors of the system in both the strong coupling regime and the weak coupling regime, respectively. In an equilibrium bath, we find that the entanglement decreases with the bath temperature and energy detuning increasing under the strong coupling regime. In the weak coupling regime,the entanglement increases with coupling strength increasing and decreases with the bath temperature and energy detuning increasing. In a nonequilibrium bath, the entanglement without RWA is useful for entanglement at lower temperatures.We also study the heat currents of the two coupled qubits and their variations with the energy detuning, coupling strength and low temperature. In the strong(weak) coupling regime, the heat current increases(decreases) with coupling strength increasing when the temperature of one bath is lower(higher) than the other, and the energy detuning leads to a positive(negative) effect when the temperature is low(high). In the weak coupling regime, the variation trend of heat current is opposite to that of coupling strength for the IHB case and the CHB case.展开更多
Until now, most researches into the rogue-wave-structure interaction have relied on experimental measurement and numerical simulation. Owing to the complexity of the physical mechanism of rogue waves, theoretical stud...Until now, most researches into the rogue-wave-structure interaction have relied on experimental measurement and numerical simulation. Owing to the complexity of the physical mechanism of rogue waves, theoretical study on the wave-structure issue still makes little progress. In this paper, the rogue wave flow around a vertical cylinder is analytically studied within the scope of the potential theory. The rogue wave is modeled by the Gauss envelope, which is one particular case of the well-known focusing theory. The formulae of the wave-induced horizontal force and bending moment are proposed. For the convenience of engineering application, the derived formulae are simplified appropriately, and verified against numerical results. In addition, the influence of wave parameters, such as the energy focusing degree and the wave focusing position, is thoroughly investigated.展开更多
A method of solving an ultracold trapped ion at the node of the standing wave laser without rotating wave approximation is proposed and the analytical forms of the eigenfunctions and eigenenergies of the system are pr...A method of solving an ultracold trapped ion at the node of the standing wave laser without rotating wave approximation is proposed and the analytical forms of the eigenfunctions and eigenenergies of the system are presented.展开更多
Considering two identical two-level atoms interacting with a single-model dissipative coherent cavity field without rotating wave approximation, we explore the entanglement dynamics of the two atoms prepared in differ...Considering two identical two-level atoms interacting with a single-model dissipative coherent cavity field without rotating wave approximation, we explore the entanglement dynamics of the two atoms prepared in different states using concurrence. Interestingly, our results show that the entanglement between the two atoms that initially disentangled will come up to a large constant rapidly, and then keeps steady in the following time or always has its maximum when prepared in some special Bell states. The model considered in this study is a good candidate for quantum information processing especially for quantum computation as steady high-degree atomic entanglement resource obtained in dissipative cavit.展开更多
We implement a binary collision approximation to study solitary wave propagation in a two-dimensional double Y- shaped granular chain. The solitary wave was transmitted and reflected when it met the interface of the b...We implement a binary collision approximation to study solitary wave propagation in a two-dimensional double Y- shaped granular chain. The solitary wave was transmitted and reflected when it met the interface of the bifurcated branches of the Y-shaped granular chains. We obtain the analytic results of the ratios of the transmitted and reflected speeds to the incident speed of the solitary wave, the maximum force between the two neighbor beads in a solitary wave, and the total time taken by the pulse to pass through each branch. All of the analytic results are in good agreement with the experimental observations from Daraio et al. [Phys. Rev. E 82 036603 (2010)]. Moreover, we also discuss the delay effects on the arrival of split pulses, and predict the recombination of the split waves traveling in branches in the final stem of asymmetric systems. The prediction of pulse recombination is verified by our numerical results.展开更多
We consider the problem of a ship advancing in waves. In this method, the zone of free surface in the vicinity of body is discretized. On the discretized surface, the first-order and second-order derivatives of ship w...We consider the problem of a ship advancing in waves. In this method, the zone of free surface in the vicinity of body is discretized. On the discretized surface, the first-order and second-order derivatives of ship waves are represented by the B-Spline formulae. Different ship waves are approximated by cubic B-spline and the first and second order derivates of incident waves are calculated and compared with analytical value. It approves that this numerical method has sufficient accuracy and can be also applied to approximate the velocity potential on the free surface.展开更多
In the limit of weak coupling between a system and its reservoir,we derive the time-convolutionless(TCL) nonMarkovian master equation for a two-level system interacting with a zero-temperature structured environment...In the limit of weak coupling between a system and its reservoir,we derive the time-convolutionless(TCL) nonMarkovian master equation for a two-level system interacting with a zero-temperature structured environment with no rotating wave approximation(NRWA).By comparing the dynamics with RWA,we demonstrate the impact of RWA on the system dynamics,as well as the effects of non-Markovianity on the preservation of atomic coherence,squeezing,and entanglement.展开更多
In this note, I introduce a mysterious approximation called the rotating wave approximation (RWA) to undergraduates or non-experts who are interested in both Mathematics and Quantum Optics. In Quantum Optics, it plays...In this note, I introduce a mysterious approximation called the rotating wave approximation (RWA) to undergraduates or non-experts who are interested in both Mathematics and Quantum Optics. In Quantum Optics, it plays a very important role in order to obtain an analytic approximate solution of some SchrÖdinger equation, while it is curious from the mathematical point of view. I explain it carefully with two coherent oscillations for them and expect that they will overcome the problem in the near future.展开更多
An efficient approximate scheme is presented for wave-propagation simulation in piecewise heterogeneous media by applying the Born-series approximation to volume-scattering waves. The numerical scheme is tested for di...An efficient approximate scheme is presented for wave-propagation simulation in piecewise heterogeneous media by applying the Born-series approximation to volume-scattering waves. The numerical scheme is tested for dimensionless frequency responses to a heterogeneous alluvial valley where the velocity is perturbed randomly in the range of 5 %–25 %,compared with the full-waveform numerical solution. Then,the scheme is extended to a heterogeneous multilayered model by calculating synthetic seismograms to evaluate approximation accuracies Numerical experiments indicate that the convergence rate of this method decreases gradually with increasing velocity perturbations. The method has a fast convergence for velocity perturbations less than 15 %. However,the convergence becomes slow drastically when the velocity perturbation increases to 20 %. The method can hardly converge for the velocity perturbation up to 25 %.展开更多
The entanglement property of two identical atoms, initially entangled in Bell states, coupled to a single-mode cavity is considered. Based on the reduced non-perturbative quantum master equation method, the entangleme...The entanglement property of two identical atoms, initially entangled in Bell states, coupled to a single-mode cavity is considered. Based on the reduced non-perturbative quantum master equation method, the entanglement evolution of the two atoms with decay is investigated beyond the conventional rotating-wave approximation. We show that the counter-rotating wave terms, usually neglected, have a great influence on the disentanglement behaviour of the system. The phenomena of entanglement sudden death and entanglement sudden birth will occur. In addition, we show that the entanglement can be strengthened by introducing the dipole-dipole interaction of the two atoms.展开更多
In this work we investigated the geometric phases of a qubit-oscillator system beyond the conventional rotating- wave approximation. We find that in the limiting of weak coupling the results coincide with that obtaine...In this work we investigated the geometric phases of a qubit-oscillator system beyond the conventional rotating- wave approximation. We find that in the limiting of weak coupling the results coincide with that obtained under rotating-wave approximation while there exists an increasing difference with the increase of coupling constant. It was shown that the geometric phase is symmetric with respect to the sign of the detuning of the quantized field from the one-photon resonance under the conventional rotating-wave approximation while a red-blue detuning asymmetry occurs beyond the conventional rotating-wave approximation.展开更多
The (e, 2e) triple differential cross sections of 2s orbitals of neon and neonic ions (Z = 11-14) are calculated using a distorted-wave Born approximation under coplanar asymmetric geometry. The calculated results...The (e, 2e) triple differential cross sections of 2s orbitals of neon and neonic ions (Z = 11-14) are calculated using a distorted-wave Born approximation under coplanar asymmetric geometry. The calculated results show that, with the increase in the nuclear charge number Z, the amplitude of triple differential cross sections decreases. The angle difference between the binary peak position and the direction of momentum transfer gradually increases with the increase in the nuclear charge Z, and a new structure appears at an ejected angle 90° 〈 θ2 〈 120°. Three kinds of collision processes are proposed to illustrate the formation mechanism of such collision peaks.展开更多
This paper considers the optimal trajectory tracking control problem for near-surface autonomous underwater vehicles(AUVs) in the presence of wave disturbances. An approximate optimal tracking control(AOTC) approach i...This paper considers the optimal trajectory tracking control problem for near-surface autonomous underwater vehicles(AUVs) in the presence of wave disturbances. An approximate optimal tracking control(AOTC) approach is proposed. Firstly, a six-degrees-of-freedom(six-DOF) AUV model with its body-fixed coordinate system is decoupled and simplified and then a nonlinear control model of AUVs in the vertical plane is given. Also, an exosystem model of wave disturbances is constructed based on Hirom approximation formula. Secondly, the time-parameterized desired trajectory which is tracked by the AUV's system is represented by the exosystem. Then, the coupled two-point boundary value(TPBV) problem of optimal tracking control for AUVs is derived from the theory of quadratic optimal control. By using a recently developed successive approximation approach to construct sequences, the coupled TPBV problem is transformed into a problem of solving two decoupled linear differential sequences of state vectors and adjoint vectors. By iteratively solving the two equation sequences, the AOTC law is obtained, which consists of a nonlinear optimal feedback item, an expected output tracking item, a feedforward disturbances rejection item, and a nonlinear compensatory term. Furthermore, a wave disturbances observer model is designed in order to solve the physically realizable problem. Simulation is carried out by using the Remote Environmental Unit(REMUS) AUV model to demonstrate the effectiveness of the proposed algorithm.展开更多
基金supported by the General Program of the National Natural Science Foundation of China(Grant No.52074295)the Special Fund for Basic Scientific Research Business Expenses of Central Universities(Grant No.2022YJSSB06)supported by State Key Laboratory for Geomechanics and Deep Underground Engineering,China University of Mining and technology,Beijing,China(Grant No.SKLGDUEK202217).
文摘A nuclear explosion in the rock mass medium can produce strong shock waves,seismic shocks,and other destructive effects,which can cause extreme damage to the underground protection infrastructures.With the increase in nuclear explosion power,underground protection engineering enabled by explosion-proof impact theory and technology ushered in a new challenge.This paper proposes to simulate nuclear explosion tests with on-site chemical explosion tests in the form of multi-hole explosions.First,the mechanism of using multi-hole simultaneous blasting to simulate a nuclear explosion to generate approximate plane waves was analyzed.The plane pressure curve at the vault of the underground protective tunnel under the action of the multi-hole simultaneous blasting was then obtained using the impact test in the rock mass at the site.According to the peak pressure at the vault plane,it was divided into three regions:the stress superposition region,the superposition region after surface reflection,and the approximate plane stress wave zone.A numerical simulation approach was developed using PFC and FLAC to study the peak particle velocity in the surrounding rock of the underground protective cave under the action of multi-hole blasting.The time-history curves of pressure and peak pressure partition obtained by the on-site multi-hole simultaneous blasting test and numerical simulation were compared and analyzed,to verify the correctness and rationality of the formation of an approximate plane wave in the simulated nuclear explosion.This comparison and analysis also provided a theoretical foundation and some research ideas for the ensuing study on the impact of a nuclear explosion.
基金the National Key Research and Development Program of China(Grant Nos.2017YFA0304202 and 2017YFA0205700)the National Natural Science Foundation of China(Grant Nos.11875231 and 11935012)the Fundamental Research Funds for the Central Universities(Grant No.2018FZA3005).
文摘We investigate the rotating wave approximation applied in the high-spin quantum system driven by a linearly polarized alternating magnetic field in the presence of quadrupole interactions.The conventional way to apply the rotating wave approximation in a driven high-spin system is to assume the dynamics being restricted in the reduced Hilbert space.However,when the driving strength is relatively strong or the driving is off resonant,the leakage from the target resonance subspace cannot be neglected for a multi-level quantum system.We propose the correct formalism to apply the rotating wave approximation in the full Hilbert space by taking this leakage into account.By estimating the operator fidelity of the time propagator,our formalism applied in the full Hilbert space unambiguously manifests great advantages over the conventional method applied in the reduced Hilbert space.
基金Project supported by the National Natural Science Foundation of China(Grant Nos.61675115 and 11704221)
文摘We study the steady-state entanglement and heat current of two coupled qubits, in which two qubits are connected with two independent heat baths(IHBs) or two common heat baths(CHBs). We construct the master equation in the eigenstate representation of two coupled qubits to describe the dynamics of the total system and derive the solutions in the steadystate with stronger coupling regime between two qubits than qubit–baths. We do not make the rotating wave approximation(RWA) for the qubit–qubit interaction, and so we are able to investigate the behaviors of the system in both the strong coupling regime and the weak coupling regime, respectively. In an equilibrium bath, we find that the entanglement decreases with the bath temperature and energy detuning increasing under the strong coupling regime. In the weak coupling regime,the entanglement increases with coupling strength increasing and decreases with the bath temperature and energy detuning increasing. In a nonequilibrium bath, the entanglement without RWA is useful for entanglement at lower temperatures.We also study the heat currents of the two coupled qubits and their variations with the energy detuning, coupling strength and low temperature. In the strong(weak) coupling regime, the heat current increases(decreases) with coupling strength increasing when the temperature of one bath is lower(higher) than the other, and the energy detuning leads to a positive(negative) effect when the temperature is low(high). In the weak coupling regime, the variation trend of heat current is opposite to that of coupling strength for the IHB case and the CHB case.
基金financially supported by the National Natural Science Foundation of China(Grant Nos.51579146,51490674,and51609101)the Shanghai Rising-Star Program(Grant No.16QA1402300)
文摘Until now, most researches into the rogue-wave-structure interaction have relied on experimental measurement and numerical simulation. Owing to the complexity of the physical mechanism of rogue waves, theoretical study on the wave-structure issue still makes little progress. In this paper, the rogue wave flow around a vertical cylinder is analytically studied within the scope of the potential theory. The rogue wave is modeled by the Gauss envelope, which is one particular case of the well-known focusing theory. The formulae of the wave-induced horizontal force and bending moment are proposed. For the convenience of engineering application, the derived formulae are simplified appropriately, and verified against numerical results. In addition, the influence of wave parameters, such as the energy focusing degree and the wave focusing position, is thoroughly investigated.
文摘A method of solving an ultracold trapped ion at the node of the standing wave laser without rotating wave approximation is proposed and the analytical forms of the eigenfunctions and eigenenergies of the system are presented.
基金The project supported by National Natural Science Foundation of China under Grant Nos. 10474118 and 10274093, the National Fundamental Research Program of China under Grant No. 2005CB724502, and the Foundation from Educational Department of Sichuan Province of China under Grant No. 2004C017
基金Project supported by the National Natural Science Foundation of China (Grant No. 10374025)the Education Ministry of Hunan Province,China (Grant No. 06A038)the Natural Science Foundation of Hunan Province,China (Grant No. 07JJ3013
文摘Considering two identical two-level atoms interacting with a single-model dissipative coherent cavity field without rotating wave approximation, we explore the entanglement dynamics of the two atoms prepared in different states using concurrence. Interestingly, our results show that the entanglement between the two atoms that initially disentangled will come up to a large constant rapidly, and then keeps steady in the following time or always has its maximum when prepared in some special Bell states. The model considered in this study is a good candidate for quantum information processing especially for quantum computation as steady high-degree atomic entanglement resource obtained in dissipative cavit.
基金Supported by the Natural Science Foundation of Hunan Province under Grant No.09JJ6011the Natural Science Foundation of Education Department of Hunan Province under Grant Nos.10A100 and 07C528
基金The project supported by National Natural Science Foundation of China under Grant Nos. 10575040, 90503010, 60478029, and 10634060, and by the State Key Basic Research Program under Grant No. 2005CB724508
基金Project supported by the Science Fund for Creative Research Groups of the National Natural Science Foundation of China (Grant No. 50921002)the Fundamental Research Funds for the Central Universities of China (Grant No. 2010LKWL09)
文摘We implement a binary collision approximation to study solitary wave propagation in a two-dimensional double Y- shaped granular chain. The solitary wave was transmitted and reflected when it met the interface of the bifurcated branches of the Y-shaped granular chains. We obtain the analytic results of the ratios of the transmitted and reflected speeds to the incident speed of the solitary wave, the maximum force between the two neighbor beads in a solitary wave, and the total time taken by the pulse to pass through each branch. All of the analytic results are in good agreement with the experimental observations from Daraio et al. [Phys. Rev. E 82 036603 (2010)]. Moreover, we also discuss the delay effects on the arrival of split pulses, and predict the recombination of the split waves traveling in branches in the final stem of asymmetric systems. The prediction of pulse recombination is verified by our numerical results.
文摘We consider the problem of a ship advancing in waves. In this method, the zone of free surface in the vicinity of body is discretized. On the discretized surface, the first-order and second-order derivatives of ship waves are represented by the B-Spline formulae. Different ship waves are approximated by cubic B-spline and the first and second order derivates of incident waves are calculated and compared with analytical value. It approves that this numerical method has sufficient accuracy and can be also applied to approximate the velocity potential on the free surface.
基金Project supported by the National Natural Science Foundation of China (Grant Nos. 11275064 and 11075050)the National Basic Research Program of China(Grant No. 2007CB925204)the Construct Program of the National Key Discipline,China
文摘In the limit of weak coupling between a system and its reservoir,we derive the time-convolutionless(TCL) nonMarkovian master equation for a two-level system interacting with a zero-temperature structured environment with no rotating wave approximation(NRWA).By comparing the dynamics with RWA,we demonstrate the impact of RWA on the system dynamics,as well as the effects of non-Markovianity on the preservation of atomic coherence,squeezing,and entanglement.
文摘In this note, I introduce a mysterious approximation called the rotating wave approximation (RWA) to undergraduates or non-experts who are interested in both Mathematics and Quantum Optics. In Quantum Optics, it plays a very important role in order to obtain an analytic approximate solution of some SchrÖdinger equation, while it is curious from the mathematical point of view. I explain it carefully with two coherent oscillations for them and expect that they will overcome the problem in the near future.
基金supported by the National Natural Science Foundation of China (Grant Nos. 41204097 and 41130418)the China National Major Science and Technology Project (2011ZX05023-005-004)
文摘An efficient approximate scheme is presented for wave-propagation simulation in piecewise heterogeneous media by applying the Born-series approximation to volume-scattering waves. The numerical scheme is tested for dimensionless frequency responses to a heterogeneous alluvial valley where the velocity is perturbed randomly in the range of 5 %–25 %,compared with the full-waveform numerical solution. Then,the scheme is extended to a heterogeneous multilayered model by calculating synthetic seismograms to evaluate approximation accuracies Numerical experiments indicate that the convergence rate of this method decreases gradually with increasing velocity perturbations. The method has a fast convergence for velocity perturbations less than 15 %. However,the convergence becomes slow drastically when the velocity perturbation increases to 20 %. The method can hardly converge for the velocity perturbation up to 25 %.
基金Supported by the Program of Fujian Education Department under Grant No. JB09069the Natural Science Fund of Fujian Province under Grant No. 2009J01275
基金Project supported by the National Natural Science Foundation of China (Grant Nos 60678022 and 10704001)the Specialized Research Fund for the Doctoral Program of Higher Education (Grant No 20060357008)+2 种基金Anhui Provincial Natural Science Foundation of China (Grant No 070412060)the Key Program of the Education Department of Anhui Province of China (Grant No KJ2008A28ZC)Anhui Key Laboratory of Information Materials and Devices (Anhui University of China)
文摘The entanglement property of two identical atoms, initially entangled in Bell states, coupled to a single-mode cavity is considered. Based on the reduced non-perturbative quantum master equation method, the entanglement evolution of the two atoms with decay is investigated beyond the conventional rotating-wave approximation. We show that the counter-rotating wave terms, usually neglected, have a great influence on the disentanglement behaviour of the system. The phenomena of entanglement sudden death and entanglement sudden birth will occur. In addition, we show that the entanglement can be strengthened by introducing the dipole-dipole interaction of the two atoms.
基金supported by the National Natural Science Foundation of China (Grant Nos. 11075099, 11047167, and 11105087)the Programme of State Key Laboratory of Quantum Optics and Quantum Optics Devices (Grant No. KF201002)+1 种基金the National Fundamental Fund of Personnel Training (Grant No. J1103210)the Youth Science Foundation of Shanxi Province of China (Grant No. 2010021003-2)
文摘In this work we investigated the geometric phases of a qubit-oscillator system beyond the conventional rotating- wave approximation. We find that in the limiting of weak coupling the results coincide with that obtained under rotating-wave approximation while there exists an increasing difference with the increase of coupling constant. It was shown that the geometric phase is symmetric with respect to the sign of the detuning of the quantized field from the one-photon resonance under the conventional rotating-wave approximation while a red-blue detuning asymmetry occurs beyond the conventional rotating-wave approximation.
基金Project supported by the Fundamental Research Funds for the Central University of China(Grant No.13CX02019A)
文摘The (e, 2e) triple differential cross sections of 2s orbitals of neon and neonic ions (Z = 11-14) are calculated using a distorted-wave Born approximation under coplanar asymmetric geometry. The calculated results show that, with the increase in the nuclear charge number Z, the amplitude of triple differential cross sections decreases. The angle difference between the binary peak position and the direction of momentum transfer gradually increases with the increase in the nuclear charge Z, and a new structure appears at an ejected angle 90° 〈 θ2 〈 120°. Three kinds of collision processes are proposed to illustrate the formation mechanism of such collision peaks.
基金supported in part by the National Natural Science Foundation of China (41276085)the Natural Science Foundation of Shandong Province (ZR2015FM004)
文摘This paper considers the optimal trajectory tracking control problem for near-surface autonomous underwater vehicles(AUVs) in the presence of wave disturbances. An approximate optimal tracking control(AOTC) approach is proposed. Firstly, a six-degrees-of-freedom(six-DOF) AUV model with its body-fixed coordinate system is decoupled and simplified and then a nonlinear control model of AUVs in the vertical plane is given. Also, an exosystem model of wave disturbances is constructed based on Hirom approximation formula. Secondly, the time-parameterized desired trajectory which is tracked by the AUV's system is represented by the exosystem. Then, the coupled two-point boundary value(TPBV) problem of optimal tracking control for AUVs is derived from the theory of quadratic optimal control. By using a recently developed successive approximation approach to construct sequences, the coupled TPBV problem is transformed into a problem of solving two decoupled linear differential sequences of state vectors and adjoint vectors. By iteratively solving the two equation sequences, the AOTC law is obtained, which consists of a nonlinear optimal feedback item, an expected output tracking item, a feedforward disturbances rejection item, and a nonlinear compensatory term. Furthermore, a wave disturbances observer model is designed in order to solve the physically realizable problem. Simulation is carried out by using the Remote Environmental Unit(REMUS) AUV model to demonstrate the effectiveness of the proposed algorithm.