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In this paper, three nonlinear finite difference schemes are proposed for solving a generalized nonlinear derivative Schrödinger equation which exposits the propagation of ultrashort pulse through optical fiber and has been illustrated to admit exact soliton-solutions. Two of the three schemes are two-level ones and the third scheme is a three-level one. It is proved that the two-level schemes only preserve the total mass or the total energy in the discrete sense and the three-level scheme preserves both the total mass and total energy. Furthermore, many numerical results are presented to test the conservative properties and convergence rates of the proposed schemes. Several dynamical behaviors including solitary-wave collisions and the first-order rogue wave solution are also simulated, which further illustrates the effectiveness of the proposed method for the generalized nonlinear derivative Schrödinger equation.

In this article, we consider the low Mach number limit of the compressible nematic liquid crystal flows in a 3D bounded domain. We establish the uniform estimates with respect to the Mach number for the strong solutions with large initial data in a short time interval. Consequently, we obtain the convergence of the compressible nematic liquid crystal system to the incompressible nematic liquid crystals system as the Mach number tends to zero.

In this paper, the existence and uniqueness of solutions to a coupled formally symmetric system of fractional differential equations with nonlinear p-Laplacian operator and nonlinear fractional differential-integral boundary conditions are obtained by using the matrix eigenvalue method. The Hyers–Ulam stability of the coupled formally symmetric system is also presented with certain growth conditions. By using the topological degree theory and nonlinear functional analysis methods, some sufficient conditions for the existence and uniqueness of solutions to this coupled formally symmetric system are established. Examples are provided to verify our results.

We prove the existence and uniqueness of random attractors for the p-Laplace equation driven simultaneously by non-autonomous deterministic and stochastic forcing. The nonlinearity of the equation is allowed to have a polynomial growth rate of any order which may be greater than p. We further establish the upper semicontinuity of random attractors as the intensity of noise approaches zero. In addition, we show the pathwise periodicity of random attractors when all non-autonomous deterministic forcing terms are time periodic.

In this paper, we prove the existence of global smooth solutions of the spin-polarized transport equation using energy estimates method in dimension two.

It is proved that blow-up solutions to N coupled Schrodinger equations $$ ivarphi_{jt} + varphi_{jxx} + mu_j|varphi_j|^{p-2}varphi_j +sum_{k eq j,;k=1}^Neta_{kj}|varphi_k|^{p_k}|varphi_j|^{p_j-2} varphi_j=0 $$ exist only under the condition that the initial data have strictly negative energy.

A 2-dimensional stochastic Burgers equation with dissipative term perturbed by Wiener noise is considered. The aim is to prove the well-posedness, existence, and uniqueness of invariant measure as well as strong law of large numbers and convergence to equilibrium.

By introducing a compactness lemma and considering a constrained variational problem, we obtain a set Gℝ2 of steady waves for Hasegawa-Mima equation, which describes the motion of drift waves in plasma. Moreover, we prove that Gℝ2 is a stable set for the initial value problem of the equation, in the sense that a solution ψ(t) which starts near Gℝ2 will remain near it for all time.

This paper focuses on establishing the existence of smooth solutions for the incompressible Landau-Lifshitz-Gilbert equation with Dzyaloshinskii-Moriya (DM) interaction and V flow in two-dimensional domains, including the torus and Euclidean space. The primary objective is to establish global existence results by investigating the conditions under which a smooth solution exists for all time, provided that the L² norm of the initial magnetization gradient is sufficiently small. Rigorous mathematical proofs for these global existence results are provided by combining analytical techniques and energy estimates. These findings enhance our understanding of solution behavior and regularity in the Landau-Lifshitz-Gilbert equation with DM interaction, shedding light on its dynamics in different spatial domains. [ABSTRACT FROM AUTHOR]