On multi-soliton solutions for the (2+1)-dimensional breaking soliton equation with variable coefficients in a graded-index waveguide

In this work, we construct multi-soliton solutions of the $(2+1)$-dimensional breaking soliton equation with variable coefficients by using the generalized unified method. We employ this method to obtain double- and triple-soliton solutions. Furthermore, we study the nonlinear interactions between these solutions in a graded-index waveguide. The physical insight and the movement role of the waves are discussed and analyzed graphically for different choices of the arbitrary functions in the obtained solutions. The interactions between the solitons are elastic whether the coefficients of the equation are constants or variables.     

Lump solutions to a (2+1)-dimensional extended KP equation

With the aid of a computer algebra system, we present lump solutions to a ($2+1$)-dimensional extended Kadomtsev–Petviashvili equation (eKP) and give the sufficient and necessary conditions to guarantee analyticity and rational localization of the solutions. We plot a few solutions for some specific values of the free parameters involved and finally derive one of the lump solutions of the Kadomtsev–Petviashvili (KP) equations from the lump solutions of the eKP equation.     

Interaction solutions of a (2+1)-dimensional dispersive long wave system

With the help of the consistent $tanh$ expansion, this paper obtains the interaction solutions between solitons and potential Burgers waves of a (2$+$1)-dimensional dispersive long wave system. Based on some known solutions of the potential Burgers equation, the multiple resonant soliton wave solutions, soliton–error function wave solutions, soliton–rational function wave solutions and soliton–periodic wave solutions are obtained directly.     

Lax pair,Darboux transformation and Nth-order rogue wave solutions for a (2+1)-dimensional Heisenberg ferromagnetic spin chain equation

A new (2+1)-dimensional Heisenberg ferromagnetic spin chain equation is investigated, which can be used to describe magnetic soliton excitations in two dimensional space fields and a time field. The Lax pair of the equation is first constructed. Based on the Lax pair, initial seed solution and Darboux transformation, the analytic first-, second- and third-order rogue wave solutions are obtained, and a general expression of the $N$th-order ($N>3$) rogue wave solutions is presented. The impacts of the system parameters on the rogue waves are demonstrated through numerical visualization method.     

Equivalent a posteriori error estimates for elliptic optimal control problems with L1-control cost

An elliptic optimal control problem involving the $L^1$ norm of the control in the cost functional is considered in this paper. We use the full discretization and the variational discretization to approximate the control problem and the efficient and reliable a posteriori error estimates are obtained for the two cases. For the variational discretization, we also analyze the convergence of adaptive finite element methods. In the end, some examples are provided to validate our analysis.     

C1 continuous h-adaptive least-squares spectral element method for phase-field models

To describe the interfacial dynamics between two phases using the phase-field method, the interfacial region needs to be close enough to a sharp interface so as to reproduce the correct physics. Due to the high gradients of the solution within the interfacial region and consequent high computational cost, the use of the phase-field method has been limited to the small-scale problems whose characteristic length is similar to the interfacial thickness. By using finer mesh at the interface and coarser mesh in the rest of computational domain, the phase-field methods can handle larger scale of problems with realistic interface thicknesses. In this work, a $C^1$ continuous $h$-adaptive mesh refinement technique with the least-squares spectral element method is presented. It is applied to the Navier–Stokes-Cahn–Hilliard (NSCH) system and the isothermal Navier–Stokes–Korteweg (NSK) system. Hermite polynomials are used to give global differentiability in the approximated solution, and a space–time coupled formulation and the element-by-element technique are implemented. Two refinement strategies based on the solution gradient and the local error estimators are suggested, and they are compared in two numerical examples.     

N-solitons,breathers, lumps and rogue wave solutions to a (3+1)-dimensional nonlinear evolution equation

Based on Hirota bilinear method, $N$-solitons, breathers, lumps and rogue waves as exact solutions of the (3+1)-dimensional nonlinear evolution equation are obtained. The impacts of the parameters on these solutions are analyzed. The parameters can influence and control the phase shifts, propagation directions, shapes and energies for these solutions. The single-kink soliton solution and interactions of two and three-kink soliton overtaking collisions of the Hirota bilinear equation are investigated in different planes. The breathers in three dimensions possess different dynamics in different planes. Via a long wave limit of breathers with indefinitely large periods, rogue waves are obtained and localized in time. It is shown that the rogue wave possess a growing and decaying line profile that arises from a nonconstant background and then retreat back to the same nonconstant background again. The results can be used to illustrate the interactions of water waves in shallow water. Moreover, figures are given out to show the properties of the explicit analytic solutions.     

Finite iterative Hermitian R-conjugate solutions of the generalized coupled Sylvester-conjugate matrix equations

In this paper, an iterative algorithm for solving a generalized coupled Sylvester-conjugate matrix equations over Hermitian $R$-conjugate matrices given by $A_{1}V{B}_1+C_{1}W{D}_1=E_1\overline{V}{F}_1+G_1$ and $A_{2}V{B}_2+C_{2}W{D}_2=E_2\overline{V}{F}_2+G_2$ is presented. When these two matrix equations are consistent, the convergence theorem shows that a solution can be obtained within finite iterative steps in the absence of round-off error for any initial arbitrary Hermitian $R$-conjugate solution matrices $V_1$, $W_1$. Some lemmas and theorems are stated and proved where the iterative solutions are obtained. A numerical example is given to demonstrate the behavior of the proposed method and to support the theoretical results.     

Galerkin finite element solution for Benjamin–Bona–Mahony–Burgers equation with cubic B-splines

In this article, we study solitary-wave solutions of the nonlinear Benjamin–Bona–Mahony–Burgers(BBM–Burgers) equation based on a lumped Galerkin technique using cubic B- spline finite elements for the spatial approximation. The existence and uniqueness of solutions of the Galerkin version of the solutions have been established. An accuracy analysis of the Galerkin finite element scheme for the spatial approximation has been well studied. The proposed scheme is carried out for four test problems including dispersion of single solitary wave, interaction of two, three solitary waves and development of an undular bore. Then we propose a full discrete scheme for the resulting IVP. Von Neumann theory is used to establish stability analysis of the full discrete numerical algorithm. To display applicability and durableness of the new scheme, error norms $L_2$, $L_{\infty }$ and three invariants $I_1,I_2$ and $I_3$ are computed and the acquired results are demonstrated both numerically and graphically. The obtained results specify that our new scheme ensures an apparent and an operative mathematical instrument for solving nonlinear evolution equation.     

Dichotomy of solutions to discrete p-Laplace equations and p-Laplace parabolic equations

In this paper, we discuss and answer the following dichotomy problems: Let $S$ be a network and ${\Delta }_{p,\omega }$ be a discrete $p$-Laplace operator with $1<p<\infty$.(i) If $u,v$ are functions satisfying

then either $u\equiv v$ on $\overline{S}$ or $u<v$ in $S$.(ii) If $u,v$ are functions satisfying

then either $u\equiv v$ on $\overline{S}$ or $u<v$ in $S\times (0,T)$.We believe that this work is not only interesting in itself, but also gives a clue to solve the problems defined on the continuous domain.     

Complexiton solutions for (3+1) dimensional KdV-type equation

(3$+$1) dimensional nonlinear KdV-type equation is solved  by Wazwaz and Zhaqilao’s method which is arisen by employing complex parameters instead of real parameters and considered generalization of simplified Hirota method. Complexiton solutions which include both trigonometric and exponential functions are obtained for referred equation. Also some special conditions to specify the non-singularity and type of solutions are derived.     

On the solution of coupled Stokes/Darcy model with Beavers–Joseph interface condition

In this paper, we first get the uniqueness result of the possible solution to the steady-state coupled Stokes/Darcy model with Beavers–Joseph interface condition for any physical parameters, especially for any $\alpha >0$. Then we show the existence of solutions for any $\alpha >0$ by using Galerkin method. Furthermore, we analyze the error of the corresponding coupled finite element scheme and derive the optimal error estimates.     

A variable exponent nonlocal p(x)-Laplacian equation for image restoration

In this paper, we focus on the mathematical and numerical study of a variable exponent nonlocal $p\left(X\right)$-Laplacian equation for image denoising. Based on the Semigroup Theory, we prove the existence and uniqueness of solution for the proposed model. To illustrate the efficiency and effectiveness of our model, we provide the denoising experimental results as well we compare it with some existing models in the literature.     

An iteration method for solving the linear system Ax=b

In this paper, based on a convergence splitting of the matrix $A$, we present an inner–outer iteration method for solving the linear system $Ax=b$. We analyze the overall convergence of this method without any other restriction on its parameters. Moreover, we give the accelerated inner–outer iteration method, and discuss how to apply the inner–outer iterations as a preconditioner for the Krylov subspace methods. The inner–outer iteration method can also be used for the solution of $AXB=C$. Finally, several numerical examples are given to validate the performance of our proposed algorithms.     

Notes on “The new exact solitary and multi-soliton solutions for the (2+1)-dimensional Zakharov–Kuznetsov equation”

In a recent paper referred to in the title, the author investigated a (2+1)-dimensional Zakharov–Kuznetsov (ZK) equation and claimed that with the aid of the coupled Burgers’ equations, a new multi-soliton solution were formally generated. We argue that the generated N-soliton solutions in the case of $N\ge 2$ is incorrect and is not admitted by the original (2+1)-dimensional ZK equation.     

Stochastic Landau–Lifshitz–Gilbert equation with anisotropy energy driven by pure jump noise

In this work we study a stochastic three-dimensional Landau–Lifshitz–Gilbert equation with non-zero anisotropy energy, which is drive by pure jump noise. We show existence of weak martingale solutions taking values in a two-dimensional sphere ${\mathbb{S}}^2$. The construction of the solution is based on the classical Faedo–Galerkin approximation, the compactness method and the Jakubowski version of the Skorokhod Theorem for nonmetric spaces.     

Analysis of Lie symmetries with conservation laws and solutions for the generalized (3+1)-dimensional time fractional Camassa–Holm–Kadomtsev–Petviashvili equation

In this paper, under investigated is a generalized $(3+1)$-dimensional Camassa–Holm–Kadomtsev–Petviashvili (gCH-KP) equation, which describes the role of dispersion in the formation of patterns in liquid drops. With the help of the semi-inverse method, the Euler–Lagrange equation and Agrawal’s method, the time fractional gCH-KP equation is derived in the sense of Riemann–Liouville fractional derivatives. Further, the symmetry of the $(3+1)$-dimensional time fractional gCH-KP equation is studied by fractional order symmetry. Meanwhile, based on the new conservation theorem, the conservation laws of $(3+1)$-dimensional time fractional gCH-KP equation are constructed. Finally, the solutions to the equation are given via a bilinear method and the radial basis functions (RBFs) meshless approach.     

Lump-type solutions and lump solutions for the (2+1)-dimensional generalized Bogoyavlensky–Konopelchenko equation

In this paper, a (2+1)-dimensional generalized Bogoyavlensky–Konopelchenko equation is investigated. Lump-type solutions and lump solutions are obtained with aid of symbolic computation via Hirota bilinear method and the ansatz technique. By taking the function $f$ in the Hirota bilinear form of the (2+1)-dimensional generalized Bogoyavlensky–Konopelchenko equation as the general quadratic polynomial function, a kind of lump-type solution which contains eleven parameters with six arbitrary independent parameters and two non-zero conditions is obtained. Lump solutions are found from the lump-type solutions via taking a special set of parameters, and the motion track of the lump is also described both theoretically and graphically.