scholarly journals The implementation of the unified transform to the nonlinear Schrödinger equation with periodic initial conditions

2021 ◽  
Vol 111 (1) ◽  
Author(s):  
B. Deconinck ◽  
A. S. Fokas ◽  
J. Lenells
Keyword(s):  
Evolution Equations ◽  
Initial Conditions ◽  
Boundary Value ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Boundary Values ◽  
Initial Value ◽  
Integrable Evolution ◽  
Initial Boundary Value ◽  
The Given

AbstractThe unified transform method (UTM) provides a novel approach to the analysis of initial boundary value problems for linear as well as for a particular class of nonlinear partial differential equations called integrable. If the latter equations are formulated in two dimensions (either one space and one time, or two space dimensions), the UTM expresses the solution in terms of a matrix Riemann–Hilbert (RH) problem with explicit dependence on the independent variables. For nonlinear integrable evolution equations, such as the celebrated nonlinear Schrödinger (NLS) equation, the associated jump matrices are computed in terms of the initial conditions and all boundary values. The unknown boundary values are characterized in terms of the initial datum and the given boundary conditions via the analysis of the so-called global relation. In general, this analysis involves the solution of certain nonlinear equations. In certain cases, called linearizable, it is possible to bypass this nonlinear step. In these cases, the UTM solves the given initial boundary value problem with the same level of efficiency as the well-known inverse scattering transform solves the initial value problem on the infinite line. We show here that the initial boundary value problem on a finite interval with x-periodic boundary conditions (which can alternatively be viewed as the initial value problem on a circle) belongs to the linearizable class. Indeed, by employing certain transformations of the associated RH problem and by using the global relation, the relevant jump matrices can be expressed explicitly in terms of the so-called scattering data, which are computed in terms of the initial datum. Details are given for NLS, but similar considerations are valid for other well-known integrable evolution equations, including the Korteweg–de Vries (KdV) and modified KdV equations.

Aspects of Osillations of Beams and Plates

2005 ◽  
Author(s):  
Mariya A. Zarubinska ◽  
W. T. van Horssen
Keyword(s):  
Boundary Value ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Initial Boundary Value Problems ◽  
Boundary Values ◽  
Internal Resonances ◽  
Specific Parameter ◽  
Plate Equations ◽  
Initial Boundary Value ◽  
Parameter Values

In this paper some initial boundary value problems for beam and plate equations will be studied. These initial boundary values problems can be regarded as simple models describing free oscillations of plates on elastic foundations or describing coupled torsional and vertical oscillations of a beam. An approximation for the solution of the initial-boundary value problem will be constructed by using a two-timescales perturbation method. For the plate on an elastic foundation it turns out that complicated internal resonances can occur for specific parameter values.

scholarly journals An initial-boundary value problem for Boltzmann’s non-stationary nonlinear one-dimensional four-moment system of equations

2020 ◽  
pp. 91-95
Author(s):  
G. Suleimenov
Keyword(s):  
Boundary Value ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Moment Equations ◽  
One Dimensional ◽  
The Moment ◽  
Initial Boundary Value ◽  
The Given ◽  
Dynamic Variables ◽  
Microscopic Distribution

In this article, the set of boundary conditions is defined for first and boundary value problems for the second approximation of Boltzmann’s system of one-dimensional nonlinear moment equations and their logic. For the second approximation of Boltzmann’s one-dimensional non-stationary nonlinear moment equations, which satisfies the Maxwell-Auzhan boundary condition, the theorem for the first boundary problem is considered and by proving this theorem, it is proved that there are only solutions to the given problems. It is known that in many problems of gas dynamics there is no need to describe the complete state of the gas by the function of microscopic distribution of molecules. Therefore, it is better to look for an easier way to describe the gas using macroscopic gas – dynamic variables (density, hydrodynamic average velocity, temperature) are determined in this rotations by the moments of the microscopic distribution function of the molecules, the author faced with the problem of analyzing the different moments of the Boltzmann equation. By studying the moment equations, the author obtained some information about the function of the microscopic distribution of molecules and the convergence of the moment method.

Initial-boundary value problems for linear PDEs with variable coefficients

2007 ◽  
Vol 143 (1) ◽  
pp. 221-242 ◽  
Author(s):  
P. A. TREHARNE ◽  
A. S. FOKAS
Keyword(s):  
Boundary Value Problems ◽  
Evolution Equations ◽  
Boundary Value ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Variable Coefficients ◽  
Initial Boundary Value Problems ◽  
Linear Evolution ◽  
Higher Derivatives ◽  
Initial Boundary Value

AbstractA new approach for studying initial-boundary value problems for linear partial differential equations (PDEs) with variable coefficients was introduced recently by the second author, and was applied to PDEs involving second order derivatives. Here, we extend this approach further to solve an initial-boundary value problem for a third-order evolution PDE with a space-dependent coefficient. The analysis is presented in such a way that it can be applied to PDEs with higher derivatives, and thus provides a method for solving initial-boundary value problems for a certain class of linear evolution equations with variable coefficients of arbitrary order.

Travelling waves in an initial-boundary value problem

1981 ◽  
Vol 90 (1-2) ◽  
pp. 41-61 ◽  
Author(s):  
E. J. M. Veling
Keyword(s):  
Boundary Value Problem ◽  
Asymptotic Expression ◽  
Travelling Waves ◽  
Boundary Value ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Stationary Problem ◽  
Travelling Wave ◽  
Boundary Values ◽  
Initial Boundary Value

SynopsisIn this paper we consider the initial-boundary value problem for the semihnear diffusion equation ul=uxx+f(u) on the half-line x>0, when for 0<a<1 f(0)=f(a)=f(1)=0 and f(u)<0 on (0, a), f(u)>0 on (a, 1). For a wide class of initial and boundary values a uniformly valid asymptotic expression is given to which the solution converges exponentially. This expression is composed of a travelling wave and a solution of the stationary problem.

The initial value problem for a nonlinear semi-infinite string

1978 ◽  
Vol 82 (1-2) ◽  
pp. 19-26 ◽  
Author(s):  
R. W. Dickey
Keyword(s):  
Boundary Value Problem ◽  
Classical Solution ◽  
Initial Value Problem ◽  
Boundary Value ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Initial Value ◽  
Galerkin Procedure ◽  
Initial Boundary Value ◽  
Infinite String

SynopsisThe existence of a classical solution to the initial boundary value problem for a semi-infinite extensible string is proved. The result is obtained by using a Galerkin procedure on a semi-infinite interval.

scholarly journals AN INTRODUCTION TO WELL-POSEDNESS AND FREE-EVOLUTION

2013 ◽  
Vol 28 (22n23) ◽  
pp. 1340015 ◽  
Author(s):  
DAVID HILDITCH
Keyword(s):  
Boundary Value Problem ◽  
Boundary Conditions ◽  
Initial Value Problem ◽  
Boundary Value ◽  
Fourier Method ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Well Posedness ◽  
Initial Value ◽  
Initial Boundary Value

These lecture notes accompany two classes given at the NRHEP2 school. In the first lecture I introduce the basic concepts used for analyzing well-posedness, that is the existence of a unique solution depending continuously on given data, of evolution partial differential equations. I show how strong hyperbolicity guarantees well-posedness of the initial value problem. Symmetric hyperbolic systems are shown to render the initial boundary value problem well-posed with maximally dissipative boundary conditions. I discuss the Laplace–Fourier method for analyzing the initial boundary value problem. Finally, I state how these notions extend to systems that are first-order in time and second-order in space. In the second lecture I discuss the effect that the gauge freedom of electromagnetism has on the PDE status of the initial value problem. I focus on gauge choices, strong-hyperbolicity and the construction of constraint preserving boundary conditions. I show that strongly hyperbolic pure gauges can be used to build strongly hyperbolic formulations. I examine which of these formulations is additionally symmetric hyperbolic and finally demonstrate that the system can be made boundary stable.

Global weak entropy solutions to the Euler–Poisson system with spherical symmetry

2016 ◽  
Vol 26 (09) ◽  
pp. 1689-1734
Author(s):  
Jingjing Xiao
Keyword(s):  
Spherical Symmetry ◽  
Boundary Value ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Approximate Solutions ◽  
Poisson System ◽  
Godunov Scheme ◽  
Initial Value ◽  
Weak Entropy Solution ◽  
Initial Boundary Value

In this paper, we study the initial boundary value problem for the isentropic Euler–Poisson system in an exterior domain with spherical symmetry. The initial data is supposed to be bounded and satisfy other suitable assumptions. Using a fractional step Godunov scheme, we construct the approximate solutions and prove the uniform [Formula: see text] estimates for the approximate solutions. Then the compensated compactness argument implies the convergence of the solutions. The weak entropy solution also satisfies the initial value and boundary value in the sense of trace.

On nonclassical problems for first-order evolution equations

2011 ◽  
Vol 18 (3) ◽  
pp. 441-463
Author(s):  
Gia Avalishvili ◽  
Mariam Avalishvili
Keyword(s):  
Boundary Value Problems ◽  
Parabolic Equations ◽  
Evolution Equations ◽  
Initial Conditions ◽  
Boundary Value ◽  
Initial Boundary ◽  
Initial Boundary Value Problems ◽  
First Order ◽  
Uniqueness Results ◽  
Initial Boundary Value

Abstract The present paper deals with nonclassical initial-boundary value problems for parabolic equations and systems and their generalizations in abstract spaces. Nonclassical problems with nonlocal initial conditions for an abstract first-order evolution equation with time-dependent operator are considered, the existence and uniqueness results are proved and the algorithm of approximation of nonlocal problems by a sequence of classical problems is constructed. Applications of the obtained general results to initial-boundary value problems for parabolic equations and systems are considered.

scholarly journals On the solutions of nonlinear initial-boundary value problems

2004 ◽  
Vol 2004 (5) ◽  
pp. 407-424 ◽  
Author(s):  
Vladimír Ďurikovič ◽  
Monika Ďurikovičová
Keyword(s):  
Evolution Equation ◽  
Boundary Value Problem ◽  
Boundary Value ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Initial Boundary Value Problems ◽  
Qualitative Properties ◽  
Science Models ◽  
Initial Boundary Value ◽  
The Given

We deal with the general initial-boundary value problem for a second-order nonlinear nonstationary evolution equation. The associated operator equation is studied by the Fredholm and Nemitskii operatortheory. Under local Hölder conditions for the nonlinear member, we observe quantitative and qualitative properties of the set of solutions of the given problem. These results can be applied to different mechanical and natural science models.

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