On the controllability for the Khokhlov–Zabolotskaya–Kuznetsov-like equation

Recalling the proprieties of the Khokhlov–Zabolotskaya–Kuznetsov (KZK) equation, we prove the controllability of moments result for the linear part of the KZK equation and its non-linear perturbation.     

Instantaneous blow-up of classical solutions to the Cauchy problem for the Khokhlov–Zabolotskaya equation

The Cauchy problem for a second-order nonlinear equation with mixed derivatives is considered. It is proved that its classical local-in-time solution does not exist. The blow-up of the solution is proved by applying S.I. Pohozaev and E.L. Mitidieri’s nonlinear capacity method.     

On Weak Solution of the Mixed Nonlinear Schrodinger Equations

1 Introduction We shall study the following Cauchy problem for the mixednonlinear Schrodinger equation (mixed NSE):whereα,β,γ are real constants with α≠0,U denotes the complex conjugate of U,     

Lie symmetries of a generalized Kuznetsov–Zabolotskaya–Khokhlov equation

We consider a class of generalized Kuznetsov–Zabolotskaya–Khokhlov (gKZK) equations and determine its equivalence group, which is then used to give a complete symmetry classification of this class. The infinite-dimensional symmetry is used to reduce such equations to (1+1)$(1+1)$-dimensional PDEs. Special attention is paid to group-theoretical properties of a class of generalized dispersionless KP (gdKP) or Zabolotskaya–Khokhlov equations as a subclass of gKZK equations. The conditions are determined under which a gdKP equation is invariant under a Lie algebra containing the Virasoro algebra as a subalgebra. This occurs if and only if this equation is completely integrable. A similar connection is shown to hold for generalized KP equations.     

Existence and Uniqueness of the Solution of Nonlinear Population Evolution Equations

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The Nonlinear Galerkin Methods for the Generalized Kuramoto-Sivashinsky Type Equations

We study the nonlinear Galerkin method for the following periodic initial valueproblem of a class of generalized Kuramoto-Sivashinsky type equations     

On Blowup for Time-Dependent Generalized Hartree–Fock Equations

We prove finite-time blowup for spherically symmetric and negative energy solutions of Hartree–Fock and Hartree–Fock–Bogoliubov-type equations, which describe the evolution of attractive fermionic systems (e.g. white dwarfs). Our main results are twofold: first, we extend the recent blowup result of Hainzl and Schlein (Comm. Math. Phys. 287:705–714, 2009) to Hartree–Fock equations with infinite rank solutions and a general class of Newtonian type interactions. Second, we show the existence of finite-time blowup for spherically symmetric solutions of a Hartree–Fock–Bogoliubov model, where an angular momentum cutoff is introduced. We also explain the key difficulties encountered in the full Hartree–Fock–Bogoliubov theory.     

Stochastic Nonlinear Equations of Schrödinger Type

In this article, the solution for a stochastic nonlinear equation of Schrödinger type, which is perturbed by an infinite dimensional Wiener process, is investigated. The existence of the solution is proved by using the Galerkin method. Moment estimates for the solution are also derived. Examples from physics are given in the final part of the article.     

Dispersion Analysis of Multi-symplectic Scheme for the Nonlinear Schrodinger Equations

In this paper,we study the dispersive properties of multi-symplectic discretizations for the nonlinear Schrodinger equations.The numerical dispersion relation and group velocity are investigated.It is found that the numerical dispersion relation is relevant when resolving the nonlinear Schrodinger equations.     

An Explicit Solution with Correctors for the Green–Naghdi Equations

In this paper, the water waves problem for uneven bottoms in a highly nonlinear regime is studied. It is well known that, for such regimes, a generalization of the Boussinesq equations called the Green–Naghdi equations can be derived and justified when the bottom is variable (Lannes and Bonneton in Phys Fluids 21, 2009). Moreover, the Green–Naghdi and Boussinesq equations are fully nonlinear and dispersive systems. We derive here new linear asymptotic models of the Green–Naghdi and Boussinesq equations so that they have the same accuracy as the standard equations. We solve explicitly the new linear models and numerically validate the results.     

The Chebyshev–Shamanskii Method for Solving Systems of Nonlinear Equations

We present a method, based on the Chebyshev third-order algorithm and accelerated by a Shamanskii-like process, for solving nonlinear systems of equations. We show that this new method has a quintic convergence order. We will also focus on efficiency of high-order methods and more precisely on our new Chebyshev–Shamanskii method. We also identify the optimal use of the same Jacobian in the Shamanskii process applied to the Chebyshev method. Some numerical illustrations will confirm our theoretical analysis.     

Modified Tricomi Problem for a Nonlinear System of Second Order Equations of Mixed Type

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Application of Splitting Algorithms in the Method of Finite Volumes for Numerical Solution of the Navier–Stokes Equations

We generalize the splitting algorithms proposed earlier for the construction of efficient difference schemes to the finite volume method. For numerical solution of the Euler and Navier–Stokes equations written in integral form, some implicit finite-volume predictor-corrector scheme of the second order of approximation is proposed. At the predictor stage, the introduction of various forms of splitting is considered, which makes it possible to reduce the solution of the original system to separate solution of individual equations at fractional steps and to ensure some stability margin of the algorithm as a whole. The algorithm of splitting with respect to physical processes and spatial directions is numerically tested. The properties of the algorithm are under study and we proved its effectiveness for solving two-dimensional and three-dimensional flow-around problems.     

Localization on the Boundary of Blow-up for Reaction–Diffusion Equations with Nonlinear Boundary Conditions

Abstract

In this work we analyze the existence of solutions that blow-up in finite time for a reaction–diffusion equation u _t  ? Δu = f(x, u) in a smooth domain Ω with nonlinear boundary conditions ?u/?n = g(x, u). We show that, if locally around some point of the boundary, we have f(x, u) = ?βu ^p , β ≥ 0, and g(x, u) = u ^q then, blow-up in finite time occurs if 2q > p + 1 or if 2q = p + 1 and β < q. Moreover, if we denote by T _b the blow-up time, we show that a proper continuation of the blowing up solutions are pinned to the value infinity for some time interval [T, τ] with T _b  ≤ T < τ. On the other hand, for the case f(x, u) = ?βu ^p , for all x and u, with β > 0 and p > 1, we show that blow-up occurs only on the boundary.     

The Existence of Infinitely Many Solutions for the Nonlinear Schrödinger–Maxwell Equations

In this paper, by using variational methods and critical point theory, we shall mainly be concerned with the study of the existence of infinitely many solutions for the following nonlinear Schrödinger–Maxwell equations $$\left\{\begin{array}{l@{\quad}l}-\triangle u + V(x)u + \phi u = f(x, u), \quad \; \, {\rm in} \, \mathbb{R}^{3},\\ -\triangle \phi = u^{2}, \quad \quad \qquad \quad \quad \quad \quad {\rm in} \, \mathbb{R}^{3},\end{array}\right.$$ where the potential V is allowed to be sign-changing, under some more assumptions on f, we get infinitely many solutions for the system.     

Approximation of Stochastic Nonlinear Equations of Schrödinger Type by the Splitting Method

In this article, we construct a splitting method for nonlinear stochastic equations of Schrödinger type. We approximate the solution of our problem by the sequence of solutions of two types of equations: one without stochastic integral term, but containing the Laplace operator and the other one containing only the stochastic integral term. The two types of equations are connected to each other by their initial values. We prove that the solutions of these equations both converge strongly to the solution of the Schrödinger type equation.     

Nonlinear Periodic Boundary Value Problem for Second Order Integro-differential Equations of Volterra Type

The periodic boundary value problem for a class of second order nonlinear integro-differential equations are disussed by using the monotone iterative technique. The open problem raised by Lakshmikantham in 1986 is solved.     

The Global Solution and Its Asymptotic Behaviours for one Class of System of Nonlinear Evolution Equations

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Nonmonotone Self-adaptive Levenberg–Marquardt Approach for Solving Systems of Nonlinear Equations

The well-known Levenberg–Marquardt method is used extensively to solve systems of nonlinear equations. An extension of the Levenberg–Marquardt method based on new nonmonotone technique is described. To decrease the total number of iterations, this method allows the sequence of objective function values to be nonmonotone, especially in the case where the objective function is ill-conditioned. Moreover, the parameter of Levenberg–Marquardt is produced according to the new nonmonotone strategy to use the advantages of the faster convergence of the Gauss–Newton method whenever iterates are near the optimizer, and the robustness of the steepest descent method in the case in which iterates are far away from the optimizer. The global and quadratic convergence of the proposed method is established. The results of numerical experiments are reported.