East Asian Journal on Applied Mathematics
The East Asian Journal on Applied Mathematics (EAJAM) aims at promoting study and research in Applied Mathematics in East Asia. It is the editorial policy of EAJAM to accept refereed papers in all active areas of Applied Mathematics and related Mathematical Sciences. Novel applications of Mathematics in real situations are especially welcome. Substantial survey papers on topics of exceptional interest will also be published occasionally.
Integrable mKdV Models as Reductions of AKNS Integrable Systems via Dual Similarity Transformations
This paper explores integrable modified Korteweg-de Vries (mKdV) models using dual similarity transformations. Two representative examples involving distinct similarity transformations are presented, along with their corresponding reduced Ablowitz-Kaup-Newell–Segur matrix spectral problems. These examples illustrate how reduced matrix mKdV integrable models can be systematically generated through such transformations.
Copyright (c) 2025 East Asian Journal on Applied Mathematics
Solving Schrödinger Bridge Problem via Continuous Normalizing Flow
The Schrödinger bridge problem (SBP), which can be understood as an entropy-regularized optimal transport, seeks to compute stochastic dynamic mappings connecting two given distributions. SBP has shown significant theoretical importance and broad practical potential, with applications spanning a wide range of interdisciplinary fields. While theoretical aspects of the SBP are well-understood, practical computational solutions for general cases have remained challenging. This work introduces a computational framework that leverages continuous normalizing flows and score matching methods to approximate the drift in the dynamic formulation of the SBP. The learned drift term can be used for building generative models, opening new possibilities for applications in probability flow-based methods. We also provide a rigorous $Γ$−convergence analysis for our algorithm, demonstrating that the neuron network solutions converge to the theoretical ones as the regularization parameter tends to infinity. Lastly, we validate our algorithm through numerical experiments on fundamental cases.
Piecewise Linear Maximum Entropy Method for Fredholm Integral Equations with Weakly Singular Kernel
Approximate solution of Fredholm integral equations with certain types of weakly singular algebraic kernels is obtained by the piecewise linear maximum entropy method. During implementation of the method, two-dimension numerical integration is reduced to one-dimension numerical integration, which allows to lower the computational cost. The results of numerical experiments are consistent with theoretical findings and show the effectiveness of the method.
A Reduced Order Modeling Method with Variable Separation-Based Domain Decomposition for Parametric Dynamical Systems
This paper proposes a model order reduction method for a class of parametric dynamical systems. Using a temporal Fourier transform, we reformulate these systems into complex-valued elliptic equations in the frequency domain, containing frequency variables and parameters inherited from the original model. To reduce the computational cost of the frequency-variable elliptic equations, we extend the variable-separation-based domain decomposition method to the complex-valued context, resulting in an offline-online procedure for solving the parametric dynamical systems. At the offline stage, separate representations of the solutions for the interface problem and the subproblems are constructed. At the online stage, the solutions of the parametric dynamical systems for new parameter values can be directly derived by utilizing the separate representations and implementing the inverse Fourier transform. The proposed approach is capable of being highly efficient because the online stage is independent of the spatial discretization. Finally, we present three specific instances of parametric dynamical systems to demonstrate the effectiveness of the proposed method.
A New Hybrid Trigonometric WENO Scheme for Hamilton-Jacobi Equations
In this paper, we introduce a new hybrid weighted essentially non-oscillatory (WENO) scheme grounded in trigonometric polynomials for the solution of Hamilton-Jacobi equations. This innovative approach utilizes trigonometric polynomial reconstruction as an alternative to the conventional algebraic polynomial reconstruction. Notably, the proposed scheme demonstrates reduced truncation errors in smooth regions and enhanced resolution in nonsmooth regions, outperforming the classical trigonometric WENO scheme — cf. [J. Zhu and J.X. Qiu, Commun. Comput. Phys. 8 (2010)]. In particular, we have designed a novel streamlined hybrid strategy aimed at improving the computational efficiency of the scheme, which surpasses the classical WENO scheme in this regard. A comprehensive suite of numerical experiments has been undertaken to demonstrate the merits of the proposed scheme.
A Neural Network Modeling for MHD–Radiative Natural Convection Williamson Fluid Between Concentric Cylinders
This study investigates the natural convection flow of Williamson fluid between two concentric cylinders while affected by the radiation effect and magnetic field. The inner cylinder remains fixed while the outer cylinder rotates. Additionally, magnetic field is oriented radially, which influences the flow of the fluid. Applying a proper transformation, one transform the non-linear partial differential equations of the Williamson fluid model into ordinary differential equations. Artificial neural networks (ANN) facilitate the computation of solutions to these nonlinear ordinary differential equations. Trial functions employ a multilayer perceptron neural network with tunable parameters, including weights and biases. The governing equations are satisfied by determining the trial solution’s changeable parameters by applying the Adam (adaptive moment estimation algorithm) optimization technique. Compared to the analytical solutions, the ANN’s result demonstrates good accuracy. Moreover, graphs show how pertinent parameters affect the velocity and temperature profiles. The temperature and velocity profiles get smaller as the magnetic parameter value increases. Furthermore, the temperature and velocity profiles increase as the Hall parameter value rises.
Copyright (c) 2026 East Asian Journal on Applied Mathematics
Heterogeneous Optimized Schwarz Methods for Heat Conduction in Composites with Thermal Contact Resistance
Heat transfer in composites is crucial in engineering, where imperfect layer contact induces thermal contact resistance (TCR), causing interfacial temperature jumps. We propose to solve this numerically using the optimized Schwarz method (OSM), which decouples the heterogeneous problem into homogeneous subproblems. This approach avoids ill-conditioned systems typical of monolithic methods under high contrast and interface discontinuities. Convergence of the algorithm with a standard Robin transmission condition is established via energy estimates and Fourier analysis. For accelerated convergence, a scaled Robin condition is introduced, with rigorous optimization of its free parameter. Key findings emerge due to TCR: first, larger TCR values speed up OSM convergence, achieving asymptotic mesh-independence. This contrasts with the mesh-dependent behavior observed without TCR. Second, greater heterogeneity contrast enhances convergence; third, unlike TCR-free cases, higher thermal conductivity also promotes convergence, similar to heterogeneity; finally, the scaled Robin condition outperforms the standard one in both theory and practice. Numerical tests validate the results and demonstrate the method's potential for nonlinear problems on irregular domains.
Dynamics of the Center of Mass in Arbitrary-Angle Rotating Bose-Einstein Condensates
This paper investigates the dynamics of the center of mass in three-dimensional arbitrary-angle rotating Bose-Einstein condensates (ARotBECs), which are governed by the Gross-Pitaevskii equation (GPE) with an arbitrary-angle angular momentum rotation term. The second-order ordinary differential equations (ODEs) which govern the motion of the center of mass of ARotBECs are analytically solved. Subsequently, a novel numerical scheme, which is mass conservative — i.e. the Lagrangian multiplier-based Crank-Nicolson leap-frog (LagM-CNLF) method, integrated with Fourier pseudo-spectral spatial discretization, is proposed to efficiently and accurately simulate the GPE with arbitrary-angle rotation term. Finally, distinct motion patterns of the center of mass are systematically categorized based on analytical solutions and rigorously validated through direct numerical simulations of the GPE, demonstrating robust consistency between theoretical predictions and numerical computational results. The dynamics of quantized vortices is also studied to show the effectiveness of the LagM-CNLF method.
Energy-Based Adaptive Deep Unfitted Nitsche Method for Elliptic Interface Problems
The paper proposes an energy-based adaptive sampling strategy to enhance the performance of the deep unfitted Nitsche method (DUNM) for elliptic interface problems. Instead of relying on fixed or random training points, the proposed refinement indicator dynamically concentrates samples in high-energy regions, including sharp coefficient jumps and interface singularities. This targeted allocation improves both accuracy and efficiency compared to random sampling. Numerical experiments in both two- and three-dimensional settings demonstrate that the method achieves robust accuracy and efficiency across diverse scenarios, from standard geometries to highly irregular interfaces, while effectively handling high-contrast coefficients and multi-subdomain configurations. These results confirm that the proposed adaptive strategy not only reduces training cost but also ensures reliable performance in complex interface problems.
Painlevé Analysis and Analytic Solutions of a Variable-Coefficient Sawada-Kotera System in Shallow Water, Ion-Acoustic Waves and Fluid Flow Dynamics
A variable-coefficient Sawada-Kotera system is investigated that models the nonlinear behaviors of waves in shallow water, ion-acoustic waves in plasma environments and fluid flow dynamics. The Painlevé integrability is tested by the WTC method with the simplified form of Krustal. The Hirota bilinear method is employed to derive the bilinear form. Consequently, we obtain a variety of analytic solutions, including soliton, lump, and breather solutions. In addition, the interactions between the lump soliton and one stripe soliton, among with the breather soliton and one stripe soliton are discussed.
Energy-Stable Full Discretization of the Modified Elastic Flow of Closed Curves
An energy-stable full discretization for the modified elastic flow of closed curves is proposed. This is a gradient flow of a modified elastic energy combining bending and Dirichlet energies. The minimization of Dirichlet energy can lead to improved mesh quality. Gradient flows for both isotropic and anisotropic cases are considered. We derive new evolution equations for the parameterization and curvature vector of curves in arbitrary codimension. The proposed formulation is discretized by a parametric finite element method in space and a first-order implicit scheme in time. We establish the unconditional energy stability for the fully discretized scheme. Additionally, the second-order accuracy of the BDF2 scheme is demonstrated. Numerical examples in two and three dimensions illustrate the efficiency, energy stability, and asymptotic mesh distribution of the method for simulating the modified elastic flow.
Quasi-Periodic Breathers of the Hirota-Maccari System
The Hirota-Maccari (HM) system generalizing the Hirota equation, serves as a generalized (2+1)-dimensional model in fluid dynamics, plasma physics, and optical fiber communication. In this paper, we obtain the quasi-periodic breathers to HM system by using the Hirota’s bilinear method and the theta function. Asymptotic analysis demonstrates that the quasi-periodic breathers can be reduced to regular breathers under small amplitude limits. Moreover, we also classified solutions based on their asymptotic behavior. Numerical examples are given to confirm the theoretical analysis.
High-Order BDFk Parametric Finite Element Methods for Anisotropic Surface Diffusion Flows and Applications in Solid-State Dewetting
In this paper, we extend the BGN formulation [J.W. Barrett, H. Garcke and R. Nürnberg, J. Comput. Phys. 222 (2007)] by incorporating the $k$-order backward differentiation formulae (BDFk) for time discretization. This allows us to develop high-order temporal parametric finite element methods for simulating anisotropic surface diffusion flows and solid-state dewetting problems, achieving accuracy levels from second-order to fourth-order. We prove the well-posedness of the constructed high-order schemes. The proposed schemes maintain good mesh quality characteristic of the classical first-order BGN scheme. Finally, we present several numerical simulations to demonstrate the high-order temporal accuracy and verify the preservation of good mesh quality and energy stability throughout the evolution.
A Greedy Randomized Average Block Projection Method for Linear Feasibility Problems
The randomized projection (RP) method is a simple iterative scheme for solving linear feasibility problems and has gained popularity due to its speed and low memory requirement. This paper develops an accelerated variant of the standard RP method by using two ingredients: the greedy probability criterion and the average block approach, and obtains a greedy randomized average block projection (GRABP) method for solving large-scale systems of linear inequalities. We demonstrate that the GRABP method achieves deterministic linear convergence with various extrapolated step sizes. Numerical experiments on both randomly generated and real-world data show the advantage of GRABP over several state-of-the-art solvers, such as the RP method, the sampling Kaczmarz Motzkin (SKM) method, the generalized SKM method, and the Nesterov acceleration of SKM method.
An Energy Stable TFPM-Based Petrov-Galerkin Scheme for Solving the Allen-Cahn Equation
An energy stable tailored finite point method based Petrov-Galerkin scheme to solve Allen-Cahn equation is proposed. In time discretization, we present both first-order and second-order semi-discrete schemes based on stabilized and convex-splitting techniques, which satisfy unconditional energy stability. We prove the maximum bound preserving principle for first-order schemes. Due to nonlinearity, the well-posedness of weak formulations based on semi-discrete schemes are demonstrated. As the nature of singularly perturbation in semi-discrete level remains when $ε$ is extremely small, we establish a specified Petrov-Galerkin scheme which leads to a unified way for space discretization. To this end, we set up nonlinear solvers which are proved to be stable and convergent. Then we construct our Petrov-Galerkin scheme, which is built upon problem-dependent test function space. The stability and second-order convergence of this scheme are rigorously proved in one dimension. In order to compute test functions, specialized TFPM schemes are incorporated into the scheme. Numerical experiments show the accuracy, efficiency, and the good performance of the method on uniform meshes even when mesh size $h$ is much larger than $ε$.
Piecewise Smooth $N$-Dimensional Nonlinear Singular Singularly Perturbed Boundary Value Problems
Internal layer phenomena are often appear in various piecewise smooth nonlinear singular singularly perturbed problems of natural sciences. To characterize these special structures, we study piecewise smooth $n$-dimensional nonlinear singular singularly perturbed boundary value problems. In particular, we show the existence of solutions with an internal layer and construct their asymptotic expansions. The remainder estimations of the approximate solutions are also given. Finally, an example aimed to verify the correctness of the developed theory is presented.
Schwartz Duality for Singularly Perturbed Differential Equations with Chebyshev Spectral Methods
Singularly perturbed differential equations with the Dirac delta function usually yield discontinuous solutions. Therefore, careful consideration is required when using numerical methods to solve these equations because of the Gibbs phenomenon. A remedy based on the Schwartz duality has been previously proposed, yielding superior results without oscillations. However, this approach has primarily been applied to linear problems and still exhibits the Gibbs phenomenon when extended to nonlinear or higher-dimensional problems. In this paper, we propose a consistent yet simple approach based on Schwartz duality that can handle such problems. Our proposed approach utilizes a modified direct projection method with a consistent discrete derivative of the Heaviside function, which directly approximates the Dirac delta function. As numerical examples, we consider several problems, including the Burgers’ equation and the two-dimensional time-dependent advection equation. The proposed method effectively eliminates Gibbs oscillations without the need for traditional regularization and demonstrates uniform error reduction for the problems considered.
Differential equations with singular sources
A New Multilevel Homotopic Adaptive Finite Element Algorithm for Convection-Dominated Diffusion-Reaction Problems
The multilevel homotopic adaptive finite element method (MHAFEM) was developed for solving convection-dominated diffusion-reaction problems on triangular meshes, where the mesh adaptation (refinement and mesh smoothing/moving) is based on an optimal interpolation error estimate in $L^p$ norm. In this paper, we follow the framework of the MHAFEM to give a new multilevel homotopic adaptive finite element algorithm. This new MHAFEM executes mesh refinement based on an a posteriori error estimator instead of the interpolation error estimate. We apply this algorithm to 1D and 2D example problems using 1D linear elements and 2D quadrilateral transition elements, respectively. Numerical results demonstrate the remarkable efficiency of the algorithm.
Convection-dominated diffusion-reaction problem
multilevel homotopic adaptive finite element method
A Fully Decoupled Linear Second-Order Energy Stable Numerical Scheme for the Moving Contact Line Problem
A decoupled linear second-order accurate unconditionally energy stable scheme for a phase-field model of the moving contact line (MCL) problem is proposed. The model consists of coupled Cahn-Hilliard and Navier-Stokes equations with the generalized Navier boundary condition. The scheme proposed introduces three scalar auxiliary variables to decouple nonlinear coupling terms while retaining stability and accuracy. Two of these variables represent the bulk and boundary energy, and one captures the “zero-energy-contribution” property between convection and surface tension terms. All related terms are decoupled by applying semi-explicit treatments while preserving stability and accuracy. Further coupling between velocity and pressure is removed by adopting a projection method. The overall scheme is second-order accurate, and the unconditional stability of energy is rigorously proved. Numerical results demonstrate the accuracy, efficiency, and stability of the method for the MCL problem. Besides, a consistent implementation of the contact angle hysteresis (CAH) is conducted to model important CAH phenomena.
The Quasi-Reversibility Regularization for Source Inversion of a Linear Kuramoto-Sivashinsky Equation
The source inversion problem for the linear Kuramoto-Sivashinsky equation is studied. To address the ill-posedness of the source term inversion, a quasi-reversibility regularization method is used for the stable reconstruction of the source term. In the case of noisy measurement data, the convergence (error estimate) of regularized solutions with respect to noise level is analyzed under a priori choice of the regularization parameter. Subsequently, a modified Morozov discrepancy principle is proposed to select regularization parameters for the a posteriori principle. The convergence (error estimate) of regularized solutions is established under this a posteriori principle. Numerical examples are provided to verify the effectiveness of the proposed quasi-reversibility regularization method.
A Fast Cascadic Multigrid Method for Exponential Compact FD Discretization of Singularly Perturbed Convection-Diffusion Equations
Solving singularly perturbed convection-diffusion equations, especially for 3D problems, is a challenging problem. In this paper, we extend our work on the extrapolation cascadic multigrid (EXCMG) method for solving the 3D Poisson equation — cf. [Pan et al., J. Sci. Comput. 2017], to 3D convection-diffusion equations with singularly perturbed parameters. First, we introduce an exponential higher order compact finite difference scheme to discretize the 3D convection-diffusion equation with variable convection coefficients, resulting in a larger-scale nonsymmetric linear system. Then, we propose an EXCMG method combined with the biconjugate gradient stabilized smoother to solve the larger-scale nonsymmetric system efficiently. Numerical experiments demonstrate that the EXCMG method is a highly effective solver for convection-dominated problems, which outperforms the existing multigrid methods such as aggregation-based algebraic multigrid method.
Geometric Approach to Symmetric Positive Definite Linear Systems
This paper compares the performance of the conjugate gradient method and geometric approach in the case of symmetric positive definite (SPD) linear systems. This approach is based on the geometric theory of ODEs which was effectively initiated by Poncaré and Liapunov. The simplest and most obvious advantage of the geometric approach over the conjugate gradient method (the MATLAB code pcg) is that this approach can find the inverse of the underlying positive definite matrix and the solution. We present various numerical examples, which demonstrate the superiority of the geometric approach. For SPD linear systems, this approach provides much higher accuracy than the conjugate gradient method. In particular, since it is a one-stop procedure, it can avoid the growth of accumulated round-off errors to some extent.
Identifying the Order and a Space Source Term in a Time Fractional Diffusion-Wave Equation
This paper is devoted to identifying the order of time fractional derivative and a space-dependent source term in a time fractional diffusion-wave equation from some additional measured data in a subdomain or on a subboundary with a small time period. The Lipschitz continuity of forward operators mapping the unknown order and source term into the given data are established based on the stability estimates of solution for the direct problem. We prove the uniqueness of the considered inverse problems by using the asymptotic behavior of the solution at $t$ = 0, the Titchmarsh convolution theorem and the Duhamel principle. Moreover, a Tikhonov-type regularization method is proposed with $H^1$-norm as a penalty term. The existence of the regularized solution and its convergence to the exact solution under a suitable regularization parameter choice are obtained. Then we employ a linearized iteration algorithm combined with the piecewise linear finite element approximation to find simultaneously the approximate order and space source term. Three numerical examples for one- and two-dimensional cases are tested and the numerical results demonstrate the effectiveness of the proposed method.
Asymptotic Behavior of Solutions for a Free Boundary Problem with a Monostable Nonlinearity
A free boundary problem with Dirichlet boundary conditions is studied. Such problems can be used for describing the spread of chemical substances or biological species, which live in a moving region $[0,h(t)].$ In this case, the free boundary $h(t)$ represents the spreading front. If the density of the substance or the population at the boundary exceeds a threshold value, they will be going to spread outwards. On the other hand, the outside environment may be not very beneficial for spreading and this generates a decay rate. We mainly analyze how the decay rate and threshold value affect the solutions. There is a trichotomy result: the solution is either shrinking, or the transition case, or spreading. Besides, if the decay rate or threshold value is large, only the shrinking happens.
High-Order Operator-Compensation Schemes with Mass and Energy Conservation for the Multi-Dimensional Zakharov System
This paper presents high-order operator-compensation (OC) methods and the Crank-Nicolson (CN) technology to solve the Zakharov system (ZS). Firstly, the OC method is used for spatial discretization, and semi-discrete high-order OC schemes are obtained, which can maintain mass and energy conservation. Subsequently, the CN technology is adopted for temporal discretization, and full-discrete high-order OC schemes are obtained, which can inherit mass and energy conservation and achieve arbitrary even-order accuracy in space and second-order accuracy in time. In the process of solving ZS, we employ the linearization method to handle the nonlinear ZS, which significantly improves the computational efficiency. Considering the large stencil of numerical schemes, we propose a fast numerical iterative method to deal with the problem, which saves the computational cost. Numerical experiments are given to test accuracy order, verify conservation property, and simulate dynamic evolution.
Exact Solutions for a Higher-Order Boussinesq Equation and a New Higher-Order Boussinesq-Like Equation
A kind of higher-order Boussinesq equation is studied in this work. Based on the Bell polynomials theories, bilinear representations of the equation are derived, and diverse interaction solutions are constructed through Hirota’s bilinear method. These include interaction solutions characterized by hyperbolic cosine and cosine functions, as well as interactions between lump and soliton solutions. In addition, generalized bilinear operators are used in order to construct a new higher-order Boussinesq-like equation, while lump and breather solutions are also developed utilizing Hirota’s bilinear technique. For the various explicit solutions obtained in this work, several of them are considered to selected particular values for the relevant parameters in order to plot different kind of three-dimensional surfaces with associated two-dimensional density profiles to give a comprehensive understanding of the evolution for various solutions.
Improved Random Feature Method for Continuous Solution Reconstruction from Sparse Observation
The reconstruction of continuous solutions using all available mechanisms and data is essential for high-precision simulations and forecasts. This paper presents an improved random feature method (IRFM) that combines observational data with models for solution reconstruction. For the multi-region neuron approximation, we employ a global activation function with robust local approximation capabilities instead of the traditional piecewise method. This enhances smoothness across regions and accelerates convergence. We also derived the equivalence condition for the optimal solution of multi-constrained optimization problems, established criteria for determining the weights in the cost function and the distribution of randomly generated collocation points, reducing biases from subjective choices. Additionally, we introduce a weighted scheme for computing the cost function related to sparse observations, reducing interpolation errors and improving stability against noise. Numerical examples demonstrate that the IRFM is more accurate and converges faster than the original RFM. Its efficiency and accuracy are validated through comparisons with physics-informed neural networks, and its flexibility is shown by successful continuous solution reconstruction in complex domains.
A Fast Discontinuous Galerkin Finite Element Method for a Bond-Based Linear Peridynamic Model
A fast discontinuous Galerkin finite element algorithm on a non-uniform mesh for solution of a one-dimensional bond-based linear peridynamic model with fractional noise is developed. It is based on the approximation of the stiffness matrix corresponding to the discontinuous Galerkin finite element method by its hierarchical representation. The fast algorithm reduces the storage requirement for the stiffness matrix from $\mathscr{O} (N^2 )$ to $\mathscr{O} (kN),$ where $k$ is a parameter controlling the accuracy of hierarchical matrices. The computational complexities of assembling the stiffness matrix and the Krylov subspace method for solving linear systems are also reduced from $\mathscr{O} (N^2)$ to $\mathscr{O} (kN).$ Numerical results show the utility of the numerical method.
Error Analysis of $hp$ Spectral Element Approximation for Optimal Control Problems with Control Constraint
In this paper, an $hp$ spectral element approximation for distributed optimal control problem governed by an elliptic equation is investigated, whose objective functional does not include the control variable. And the constraint set on control variable is stated with $L^2$-norm. Optimality condition of the continuous and discretized systems are deduced. In order to solve the equivalent systems with high accuracy, $hp$ spectral element method is employed to discretize the constrained optimal control systems. Based on the property of some interpolation operators, a posteriori error estimates are also established by using some properties of some interpolation operators carefully. Finally, a projection gradient algorithm and a numerical example are provided, which confirm our analytical results. Such estimators guarantee the construction of reliable adaptive methods for optimal control problems.
Multi-View Data Clustering via Dynamical Optimization of Consensus Laplacian Matrix
Multi-view data analysis has gained increasing popularity, in particular multi-view spectral clustering has attracted much attention for its outstanding performance in mining heterogeneity information in multi-view data. However, most spectral clustering methods exhibit the following disadvantages: firstly, learning consensus representation directly from multi-view data that may contain noise renders a distorted description; secondly, the traditional two-step process may fall into a suboptimal solution. To overcome these disadvantages, a novel multi-view spectral clustering method is proposed by unifying the optimization of consensus Laplacian matrix and the learning and discretization of spectral embedding into one step. We consider that the optimal Laplacian matrix is in the neighborhood of view-specific Laplacian matrix, as the view-specific Laplacian matrix only contains partial information from multi-view data, resulting in certain deviation from the optimal Laplacian matrix. The consensus Laplacian matrix was obtained in a dynamic optimization way with the spectral rotation and embedding information simultaneously determined. Extensive experiments have been conducted to demonstrate the effectiveness and superiority of our proposed method.