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《Communications in Nonlinear Science & Numerical Simulation》2011,16(3):1697-1698
We analyze the paper by Wazwaz [Wazwaz AM. A study on an integrable system of coupled KdV equations. Commun Nonlinear Sci Numer Simul, 2010;15:2846–2850]. Author tried to show that the system of coupled KdV equations is completely integrable but he has used the curious approach for the proof. We demonstrate that, author has taken the relation between dependent variables and has obtained the well known result by Hirota for the KdV equation. 相似文献
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A numerical study of variable depth KdV equations and generalizations of Camassa–Holm-like equations
In this paper we numerically study the KdV-top equation and compare it with the Boussinesq equations over uneven bottoms. We use here a finite-difference scheme that conserves a discrete energy for the fully discrete scheme. We also compare this approach with the discontinuous Galerkin method. For the equations obtained in the case of stronger nonlinearities and related to the Camassa–Holm equation, we find several finite difference schemes that conserve a discrete energy for the fully discrete scheme. Because of its accuracy for the conservation of energy, our numerical scheme is also of interest even in the simple case of flat bottoms. We compare this approach with the discontinuous Galerkin method. 相似文献
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Kadomtsev–Petviashvili (KP)-type equations are seen in fluid mechanics, plasma physics, and gas dynamics. Hereby we consider an integrable KP-based system. With the Hirota method, symbolic computation and truncated Painlevé expansion, we obtain bright one- and two-soliton solutions. Figures are plotted to help us understand the dynamics of regular and resonant interactions, and we find that the regular interaction of solitons is completely elastic. Based on the asymptotic and graphical behavior of the two-soliton solutions, we analyze two kinds of resonance between the solitons, both of which are non-completely elastic. A triple structure, a periodic resonant structure in the procedure of interactions and a high wave hump in the vicinity of the crossing point, can be observed. Through the linear stability analysis, instability condition for the soliton solutions can be given, which might be useful, e.g., for the ship traffic on the surface of water. 相似文献
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Jie Wu 《Applicable analysis》2013,92(7):1224-1235
In this short note, we establish the global existence of weak solutions and classical solutions to the two-dimensional chemotaxis-Navier–Stokes system for both the Cauchy problem and the initial-boundary value problem under some suitable small conditions on the initial data. In particular, we improve the recent results obtained by Duan–Li–Xiang (J. Differential Equations, 2017). 相似文献
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In this paper, we give a probabilistic interpretation for a coupled system of Hamilton–Jacobi–Bellman equations using the
value function of a stochastic control problem. First we introduce this stochastic control problem. Then we prove that the
value function of this problem is deterministic and satisfies a (strong) dynamic programming principle. And finally, the value
function is shown to be the unique viscosity solution of the coupled system of Hamilton–Jacobi–Bellman equations. 相似文献
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Annali di Matematica Pura ed Applicata (1923 -) - We study the following system of nonlinear Schrödinger equations: $$\begin{aligned} \left\{ \begin{array}{l} -\varepsilon ^2\Delta u +a(x) u =... 相似文献
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Janete S. Carvalho Liliane A. Maia Olimpio H. Miyagaki 《Zeitschrift für Angewandte Mathematik und Physik (ZAMP)》2011,26(3):67-86
We consider the nonlinear Schr?dinger equation
-\triangle u + V(x)u = f(u) in \mathbbRN.-\triangle u + V(x)u= f(u)\quad {\rm in}\quad \mathbb{R}^N. 相似文献
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Janete S. Carvalho Liliane A. Maia Olimpio H. Miyagaki 《Zeitschrift für Angewandte Mathematik und Physik (ZAMP)》2011,62(1):67-86
We consider the nonlinear Schrödinger equation 相似文献
$$-\triangle u + V(x)u= f(u)\quad {\rm in}\quad \mathbb{R}^N.$$ 11.
In this note, we study the existence and multiplicity of solutions for a system of coupled elliptic equations. We introduce a revised Ambrosetti–Rabinowitz condition, and show that the system has a nontrivial solution or even infinitely many solutions. 相似文献
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Yurii V. Brezhnev 《Journal of Differential Equations》2012,253(12):3727-3751
We show that four exceptional Fuchsian equations, each determined by the four parabolic singularities, known as the Chudnovsky equations, are transformed into each other by algebraic transformations. We describe equivalence of these equations and their counterparts on tori. The latters are the Fuchsian equations on elliptic curves and their equivalence is characterized by transcendental transformations which are represented explicitly in terms of elliptic and theta functions. 相似文献
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This paper discusses the weakly coupled nonlinear Schrödinger equations in the supercritical case. With the best constant of the Gagliardo-Nirenberg inequality, we derive a sufficient condition for the global existence of solutions; this condition is expressed in terms of stationary solutions (nonlinear ground state). 相似文献
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C.V. Pao Yu-Hsien Chang Guo-Chin Jau 《Nonlinear Analysis: Real World Applications》2013,14(6):2152-2165
This paper is concerned with two mathematical models which describe the transient behavior of a catalytic converter in automobile engineering. The first model consists of a coupled system of a heat-conduction equation and two integral equations while the second model involves only one integral equation. It is shown that for any nonnegative initial and boundary functions the three-equation model has a unique bounded global solution while the solution of the two-equation model blows up in finite time. The proof for the global existence and finite-time blow-up property of the solution is by the method of upper and lower solutions and its associated monotone iteration. This method can be used to develop computational algorithms for numerical solutions of the coupled systems. 相似文献
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We consider the scalar problem on the diffraction of a plane wave on a system of two screens with boundary conditions of the first and the second kind and a solid inhomogeneous body in the semiclassical setting. The original boundary value problem for the Helmholtz equation is reduced to a system of singular integral equations over the body domain and the screen surfaces. We prove the equivalence of the integral and differential statements of the problem, the solvability of the system of integral equations in Sobolev spaces, and the smoothness of its solutions. To solve the integral equations approximately, we use the Bubnov-Galerkin method; we introduce basis functions on the body and the screens and prove the consistency and convergence of the numerical method. 相似文献
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Ladislav Adamec 《Applications of Mathematics》2005,50(2):93-101
In the theory of autonomous perturbations of periodic solutions of ordinary differential equations the method of the Poincaré mapping has been widely used. For the analysis of properties of this mapping in the case of two-dimensional systems, a result first obtained probably by Diliberto in 1950 is sometimes used. In the paper, this result is (partially) extended to a certain class of autonomous ordinary differential equations of higher dimension.This research was supported by Grant No. 201/99/0295 of the Grant Agency of the Czech Republic.This revised version was published online in April 2005 with a corrected missing date string. 相似文献
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We obtain existence and multiplicity results for the solutions of a class of coupled semilinear bi-harmonic Schrödinger equations. Actually, using the classical Mountain Pass Theorem and minimization techniques, we prove the existence of critical points of the associated functional constrained on the Nehari manifold.Furthermore, we show that using the so-called fibering method and the Lusternik–Schnirel’man theory there exist infinitely many solutions, actually a countable family of critical points, for such a semilinear bi-harmonic Schrödinger system under study in this work. 相似文献
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《中国科学 数学(英文版)》2015,(8)
Consider the following system of double coupled Schr¨odinger equations arising from Bose-Einstein condensates etc.,-△u+u=μ1u3+βuv2-κv,-△v+v=μ2v3+βu2v-κu,u≠0,v≠0 and u,v∈H1(RN),whereμ1,μ2are positive and fixed;κandβare linear and nonlinear coupling parameters respectively.We first use critical point theory and Liouville type theorem to prove some existence and nonexistence results on the positive solutions of this system.Then using the positive and non-degenerate solution to the scalar equation-△ω+ω=ω3,ω∈H1r(RN),we construct a synchronized solution branch to prove that forβin certain range and fixed,there exist a series of bifurcations in product space R×H1r(RN)×H1r(RN)with parameter κ. 相似文献
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