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In this paper, we study the Cauchy problem of the Cahn–Hilliard equation, and first reveal that the complicated asymptotic behavior of solutions can happen in high-order parabolic equation.

In this paper we study the following nonlinear Hamiltonian elliptic system with gradient term − Δ u + b → ( x ) ⋅ ∇ u + V ( x ) u = H v ( x , u , v ) i n R N , − Δ v − b → ( x ) ⋅ ∇ v + V ( x ) v = H u ( x , u , v ) i n R N . Under a local super-quadratic condition on the nonlinearity, we obtain a new existence result of nontrivial solutions by using variational method. Here we do not need the global super-quadratic condition, and in this case our result allows the nonlinearity to be super-quadratic at some domains and asymptotically quadratic at other domains. In the proofs we apply some special techniques to demonstrate the link geometry of the energy functional.

In this paper, we consider a nonlinear hyperbolic equation. We establish a blow up result for the initial and boundary value problem. Furthermore, the upper bound of the blow up time is derived.

In this work, we investigate a cooperative parabolic system u t − Δ u = u ( λ − u + b v ) , v t − Δ v = v ( μ − v + c u ) with blow-up initial and boundary values over a smooth bounded domain Ω of R N with N ≥ 1 . We show existence, nonexistence and uniqueness of solutions. We also provide an exact estimate of the behavior of the solutions near the parabolic boundary.

By using the Hirota’s bilinear method, the bilinear form of the sixth-order Ramani equation is succinctly obtained. With the aid of the obtained bilinear form, some new interaction solutions which include interaction solutions between exponential and trigonometric functions, interaction solutions between exponential and hyperbolic functions, and interaction solutions between trigonometric and hyperbolic functions are also presented by employing the three wave method.

We consider the Cauchy problem for the semilinear pseudo-parabolic equation with initial data with compact support. We find the life span of the solution, when the power exponent p is smaller than the Fujita exponent.

We present a linear, second order, energy and entropy-production-rate preserving scheme for a thermodynamically consistent phase field model for dentritic crystal growth, combining an energy quadratization strategy with the finite element method. The scheme can be decomposed into a series of Poisson equations for efficient numerical implementations. Numerical tests are carried out to verify the accuracy of the scheme and simulations are conducted to demonstrate the effectiveness of the scheme on benchmark examples.

Superpositions of hierarchies of integrable equations are also integrable. The superposed equations, such as the Hirota equations in the AKNS hierarchy, cannot be considered as new integrable equations. Furthermore if one applies the Hirota bilinear method to these equations one obtains the same $N$-soliton solutions of the generating equation which differ only by the dispersion relations. Similar discussions can be made for the locally and nonlocally reduced equations as well. We give, as an example, AKNS system of equations in $(1+1)$-dimensions., 7 pages

A fast mass-conserving explicit splitting method is proposed to solve the stochastic space-fractional nonlinear Schrodinger equation which includes a nonlinear term, a fractional Laplacian term and a multiplicative noise term resulting from the nonlocal property, the random variation of the media and the exterior random disturbance. The method splits the original equation into two sub-equations so that appropriate numerical methods can be applied to each sub-equation. A variety of numerical examples in both one- and multi-dimensional spaces show that the scheme has good mass conservative property, first-order strong convergence in time and high efficiency.

This paper studies the decay estimate of solutions to the 3D generalized magnetohydrodynamic system in the critical space χ 1 − 2 α ( R 3 ) with 1 2 ≤ α ≤ 1 . We obtain the temporal decay rate ( 1 + t ) − ( 5 4 α − 1 ) for a small initial data global solution which was established by Ye (2016) and improve the temporal decay rate.

In this paper we consider an inverse coefficients problem for a quasilinear elliptic equation of divergence form $\nabla\cdot\vec{C}(x,\nabla u(x))=0$, in a bounded smooth domain $\Omega$. We assume that $\overrightarrow{C}(x,\vec{p})=\gamma(x)\vec{p}+\vec{b}(x)|\vec{p}|^2+\mathcal{O}(|\vec{p}|^3)$, by expanding $\overrightarrow{C}(x,\vec{p})$ around $\vec{p}=0$. We give a reconstruction method for $\gamma$ and $\vec{b}$ from the Dirichlet to Neumann map defined on $\partial\Omega$., Comment: NA

In this paper, two meshless schemes are proposed for solving Dirichlet boundary optimal control problems governed by elliptic equations. The first scheme uses radial basis function collocation method (RBF-CM) for both state equation and adjoint state equation, while the second scheme employs the method of fundamental solution (MFS) for the state equation when it has a zero source term, and RBF-CM for the adjoint state equation. Numerical examples are provided to validate the efficiency of the proposed schemes.

The analysis of the existence of solutions for the systems of polynomial equations is much more difficult to handle in general. Nonetheless, we can still confirm the existence of some particular solutions, especially for some special structured tensor equations. In this paper, we theoretically prove that a system of H + -tensor equations with the positive right-hand side has a unique positive solution by proposing a positive increasing map. We also prove that this system also has a negative solution. Numerical examples are presented to verify our results are true.

Based on the robust inverse scattering method, the high-order rogue wave of generalized nonlinear Schrodinger equation with nonzero boundary is given. Using this method, we only need the elementary Darboux transformation but not with the limit progress, which is more convenient than before. By choosing different parameters c 1 and c 2 appeared in the Darboux matrix, the 2 n and 2 n − 1 order rogue waves are derived respectively. Furthermore, the general breather is also given with a different spectral parameters.

Two novel and efficient modified techniques based on recently developed stabilized invariant energy quadratization (IEQ) approach to deal with nonlinear terms in gradient flows are proposed in this paper. We proved the unconditional energy stability for a class of gradient flows and their semi-discrete schemes carefully and rigorously. One of the contributions for this approach is that we succeeded in finding suitable positive preserving functions in square root and do not need to add a positive constant which cannot be fixed before computing. Secondly, all nonlinear terms can be treated semi-explicitly, and one only needs to solve three decoupled linear equations with constant coefficients at each time step. Finally, several numerical simulations are demonstrated to verify the accuracy and efficiency of our proposed schemes.

We investigate the connection between the sign of Δ 1 − μ + a ν Δ a μ f ( t ) and the monotone behavior of t ↦ f ( t ) . In particular, given a function f : N a → R we consider the homotopy G ν : R × [ 0 , 1 ] → R defined by G ν ( f ( a ) , η ) ≔ η ⋅ 1 2 ( 1 − ν ) ( ν ) f ( a ) , and we analyze the effect of changing η ∈ [ 0 , 1 ] in the condition Δ 1 − μ + a ν Δ a μ f ( t ) ≥ G ν ( f ( a ) , η ) . We show that as η → 1 − there is an increasingly strong connection between the sign of Δ 1 − μ + a ν Δ a μ f ( t ) and the monotone behavior of f .

Based on the basic conclusions of the Riemann–Hilbert method for solving the initial value problem of the complex short-pulse equations, the general form of the two-soliton solutions of the complex short pulse equation is given in this paper. Under the two different assumptions of the scattering coefficient, the expression of the two-soliton solutions of the equation is given specifically.

We establish the energy minimality property of a solution to the generalized Painleve-II equation Δ y − x 1 y − 2 y 3 = 0 , ( x 1 , x 2 ) ∈ R 2 , which is increasing in x 2 and converges to the positive and negative Hastings–McLeod solutions as x 2 → ± ∞ .

In this paper, we investigate a fourth-order coupled nonlinear Schrodinger equation, which describes the propagation of ultrashort optical pulses in the birefringent optical fiber. Lax pair is constructed through the symbolic computation, along with the corresponding Darboux transformation (DT) which is different from that given in the existing literatures. Rogue wave solutions are obtained by virtue of the DT. Three different types of the rogue waves based on the solutions are exhibited, i.e., the anti-eye-shaped, eye-shaped four-petaled ones. Moreover, we find that the wave structure is affected by the relative background amplitudes and certain parameter in the solutions, and the range of the first-order rogue wave along the x axis increases with the increase of the fourth order term γ , while the range of the rogue wave along the t axis decreases.

In this paper, we present an efficient nonmonotone adaptive cubic regularization method with line search for solving large-scale unconstrained optimization problem. Its global convergence is analyzed. Numerical experiments are performed to show the efficiency and effectiveness of the proposed method and its superiority to the other existing methods.

In the paper, a new sufficient condition for the oscillation of all solutions of linear differential equations with general deviating argument is established. An example numerically solved in MATLAB is also given to illustrate the applicability and the strength of a newly obtained condition over the known ones.

The stability problem on perturbations near the hydrostatic equilibrium of the Boussinesq equations has recently attracted considerable attention and there are substantial developments. This paper establishes the global H 1 -stability for a 2D Boussinesq system with partial dissipation. When there is no thermal diffusion, the stability of the temperature gradient remains an open problem. This paper assesses the stability of the temperature gradient when there is a horizontal thermal diffusion.

The interest of the majorizing (scalar) sequences lies in that, from their convergence, we can deduce the convergence of an iterative method in Banach spaces. We propose a new technique to construct majorizing sequences that generalizes that given by Kantorovich for Newton’s method.

In this paper, we study the approximation system to the steady compressible Navier–Stokes equations. It is proved that the density has a positive lower bound. And we use the lower bound to study the distribution of the density for a family of elliptic equations. A further investigation gives a typical upper bound of the density, and its limit functions only fit the solutions with the velocity divergence free.

A coupled integrable lattice equation is derived from a 4 × 4 matrix spectral problem, then with the help of a special Darboux matrix, explicit solutions of the aforementioned equation are given by means of gauge transformation between the Lax pair. Finally, the density profiles of these exact solutions are presented to illustrate these solutions.

In this paper, a complex cubic–quintic Ginzburg–Landau equation (CCQGLE) is investigated. Using the asymmetric method, the analytic one-soliton solution of the CCQGLE is obtained for the first time. Through analyzing the solutions obtained, the transmission of the soliton is controlled by changing the values of related parameters. Results of this paper contribute to obtain the analytic soliton solution of the higher-order CCQGLE.

In this article, we study the Caputo type q -fractional differential equation: c D q α x ( t ) = f ( t , x ( t ) ) , 0 α , q 1 . By using the piecewise linear interpolation approximation, we present a difference formula for discretizing the Caputo type q -fractional derivative c D q α x ( t ) and derive the truncation error boundness. This difference formula is used to solve the initial value problem of q -fractional differential equation. Numerical examples are provided to verify the effectiveness of our difference method.

This paper is concerned with the long-term behavior for the strong solution of weakly dissipative wave equation with sub-quintic nonlinearity. With the help of Strichartz estimate, we can establish an energy estimate for the strong solution. Then the global attractor is obtained in ( H 2 ( Ω ) ∩ H 0 1 ( Ω ) ) × H 0 1 ( Ω ) . Furthermore, translational regularity of the attractor is established.

This paper is concerned with a chemotactic system modeling the coral broadcast spawning given by u t + U ⋅ ∇ u = Δ u − χ ∇ ⋅ ( u ∇ v ) − μ u 2 , v t + U ⋅ ∇ v = Δ v − v + u in a bounded domain Ω ⊂ R n ( n ≥ 1 ) under Neumann boundary conditions. We provide a rather simpler proof of the non-trivial bounded classical solution on the decay profile. In addition, we also obtain the optimal decay rate of ∇ v ( ⋅ , t ) in L p ( Ω ) as t → ∞ .

In this paper, we develop a new square phase-field crystal model using the L 2 -gradient flow approach, where the total mass of atoms is conserved through a nonlocal Lagrange multiplier. We construct a fast, provably unconditionally energy stable, second-order scheme by using the recently developed SAV approach with the stabilization technique, where an extra stabilization term is added to enhance the stability and keep the required accuracy while using large time steps. Through the comparisons with the classical Cahn–Hilliard type square phase-field crystal model and the non-stabilized SAV scheme for simulating some benchmark numerical examples, we demonstrate the robustness of the new model, as well as the stability and the accuracy of the developed scheme, numerically.

The Camassa–Holm equation can be used in fluids and other fields. Under investigation in this paper, the bilinear form, implicit soliton solution and multi-peakon solution of the generalized nonautonomous Camassa–Holm equation under constraints are derived. Based on these, time varying influence factors of solution amplitude, velocity and background are discussed, which are caused by inhomogeneity of boundaries and media. Furthermore, the phenomena of nonlinear tunnelling, soliton collision and split are constructed to show the characteristic of nonautonomous solitons and peakons in the propagation.

Under some assumptions we find a general solution of the factorization problem for a family of second order difference equations.

Analysing the fractional τ -norm, the uniform convergence of the V-cycle multigrid FEM for the time-dependent fractional problem is strictly proved when τ → 0 . The numerical experiments are performed to verify the convergence with O ( N log N ) complexity by fast Fourier transform method.

In this paper, we firstly develop an inequality on the rising function, which help us to deal with the fractional integral (sum) in terms of Holder inequality. Secondly, a finite-time stability criterion of a class of nonlinear fractional delay difference systems with order 1 α 2 is addressed by virtue of a new discrete Gronwall inequality and Jensen inequality. Finally, two numerical examples are presented to demonstrate our theoretical result.

This paper studies the superconvergence phenomena appearing in bubble-type adaptive finite element process. Based on a mesh condition derived from the bubble-type mesh generation, a superconvergence result of ‖ u h − u I ‖ 1 , Ω is obtained. Further, under the recovered gradient methods, a superconvergence estimation of bubble-type adaptive procedure is established. Theoretically, the reason behind the appearance of superconvergence in bubble-type adaptive procedure is justified.

In this paper we study the decay rates of the solutions for a Schrodinger equation with double power nonlinearity in L p ( R ) -norm for 2 p ≤ ∞ . We give some numerical examples to illustrate our analytical results.

In this study, we consider a diffusive predator–prey system with prey-taxis and ratio-dependent functional responses in a spatially heterogeneous environment. Prey-taxis implies that the predator exhibits directed movement in the presence of prey. We claim that when the predator diffuses uniformly without prey-taxis, only the diffusion of the species plays a role in the predator’s invasion in a spatially homogeneous environment. However, when the predator disperses through uniform diffusion together with prey-taxis, we observe that both the diffusion of species and prey-taxis affect the invasion of the predator, which is not the case in a spatially homogeneous environment. The results are obtained by investigating the local stability of a semi-trivial solution, using an eigenvalue analysis.

The aim of this article is to present the correct version of the main theorem 3.2 given in Guo and Duff (2011), concerning the semi-local convergence analysis of the Newton-HSS (NHSS) method for solving systems of nonlinear equations. Our analysis also includes the corrected upper bound on the initial point.

This paper makes a first attempt to investigate the long-time behaviour of solutions of 2D acoustic wave equation by integrating strengths of the Krylov deferred correction (KDC) method in temporal direction and the meshless generalized finite difference method (GFDM) in space domain. The KDC approach introduces the second order time derivative of acoustic pressure as a new unknown variable, and then converts the acoustic wave equation into the Helmholtz-type equation with the new variable through the temporal discretization. We finally solve the transformed equation in space domain by employing the meshless GFDM. Numerical experiments are provided to verify the accuracy of the proposed method, and numerical results illustrate the good performance of the method for long-time numerical simulations of acoustic wave problems.

Under investigation in this work is a longitudinal wave motion equation, which describes the solitary waves propagation with dispersion caused by transverse Poisson’s effect in a magneto-electro-elastic circular rod. The Lie symmetry method is employed to study its vector fields and optimal systems, respectively. Furthermore, the symmetry reductions and eight families of soliton wave solutions of the equation are obtained on the basis of the optimal systems, including hyperbolic-type and trigonometric-type solutions. Two of reduced equations are Painleve-like equations. Finally, by virtue of conservation law multiplier, the complete set of local conservation laws of the equation for the arbitrary constant coefficients is well constructed with a detailed derivation.

In this article, we obtain some properties of the positive solutions to a class of fractional differential equation multipoint boundary value problems at resonance. Necessary conditions for the existence of positive solutions are established by means of the spectral theory of linear operator. As application, necessary and sufficient conditions for the existence of positive solutions are given for the case that the nonlinearity has a fixed sign. Some examples are presented to illustrate our results.

In this paper, we present the construction of quasiperiodic solutions of the p th equation in the modified Kadomtsev–Petviashvili hierarchy by seeking compatible solutions of the 1th and the p th equations in the derivative nonlinear Schrodinger hierarchy.

In this paper, we research the geometric multiplicity of eigenvalues for a nonlocal Sturm–Liouville eigenvalue problem. To this end, we study the uniqueness of solutions for a nonlocal Sturm–Liouville problem under some initial value conditions.

This paper deals with the boundedness of nonoscillatory solutions of the forced fractional differential equation with positive and negative terms C D c α y ( t ) + f ( t , x ( t ) ) = e ( t ) + k ( t ) x ( t ) + h ( t , x ( t ) ) , where t ≥ c , α ∈ ( 0 , 1 ) , c > 1 , and C D c α y denotes the Caputo fractional derivative of y of order α . The cases where y ( t ) = a ( t ) x ′ ( t ) ′ and y ( t ) = a ( t ) x ′ ( t ) . are considered. The technique used can be applied to other related fractional differential equations. Examples are inserted to illustrate the relevance of the results obtained.

In this paper, we further consider the SOR-like iteration method for solving absolute value equations. Some new convergence conditions are obtained from the involved iteration matrix of the SOR-like iteration method, which are different from the results by Ke and Ma (2017). Numerical experiments show that the SOR-like iteration method for solving absolute value equations is efficient and feasible.

Most standard radial basis function (RBF) methods strongly depend on a shape parameter in accuracy and stability. However, the selection of optimal shape parameter in the RBF methods has long been a challenging task. In this paper, a new global RBF, known as coupled radial basis function (CRBF), is presented based on coupling the infinitely smooth RBFs with the conical spline. It is found that the Kansa’s method with the CRBF inherits the strengths of the aforementioned two types of RBFs, and achieves an accurate and stable numerical solution almost independent of the shape parameter values. Therefore, the proposed methodology completely overcome the troublesome issue of the optimal shape parameter that is a formidable obstacle to global RBF schemes. Two test examples verify the accuracy and robustness of the proposed CRBF method.

We study the conditional regularity for the incompressible Navier–Stokes equations in the whole three dimensional space in terms of one component of the velocity field u = ( u 1 , u 2 , u 3 ) . Let α ∈ ( 0 , ∞ ) . For f ∈ L 2 ( R d ) , d ∈ N , we define: ‖ f ‖ 2 , l o g α 2 = ∫ R d log α ( e + | ξ | ) | f ˆ ( ξ ) | 2 d ξ , where f ˆ denotes the Fourier transform of f , and prove the following regularity criterion: u is regular on ( 0 , T ] , T > 0 , if u 3 ∈ L ∞ ( 0 , T ; L v ∞ L h , l o g α 2 ) for some α ∈ ( 1 , ∞ ) , where v and h denote the vertical and horizontal components, respectively. This criterion possesses two substantial properties: it is almost optimal from the scaling point of view and does not almost require any information on the derivative of u 3 . The relation of the presented criterion to the kin criteria published in the literature is discussed throughout the paper.

Based on the Hirota bilinear method, three types of exact solutions of a (3+1)-dimensional nonlinear evolution equation are obtained, which contain rogue waves, breather waves and solitary waves. Moreover, the dynamical behaviors of all types of waves are discussed with graphic analysis.

In this note, we consider the blowup phenomenon of Grushin’s operator. By using the knowledge of probability, we first get an expression of heat kernel of Grushin’s operator. Then by using the properties of heat kernel and suitable auxiliary function, we get that the solution will blow up in finite time.

In this paper, shift-splitting preconditioners are studied for a special class of block three-by-three saddle point problems, which arise from many practical problems and are different from the traditional saddle point problems. It is proved that the block three-by-three saddle point matrix is positive stable and the corresponding shift-splitting stationary iteration method is unconditionally convergent, which leads to a nice clustering property of the eigenvalues of the shift-splitting preconditioned matrix. Numerical results show that the proposed shift-splitting preconditioners outperform much better than some existing block diagonal preconditioners studied recently.