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Nonlinear Schrödinger Equation and the Hyperbolization Method
Computational Mathematics and Mathematical Physics, 2022zbMATH Open Web Interface contents unavailable due to conflicting licenses.
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Oscillation Properties of Nonlinear Hyperbolic Equations
SIAM Journal on Mathematical Analysis, 1984The authors derive a number of new oscillation criteria for hyperbolic equations. First of all, three theorems are proved, giving sufficient conditions for oscillation of solutions of the characteristic initial value problem \[ (2.2)\quad u_{xy}+c(x,y,u)=f(x,y),\quad u_ x(x,0)=g(x),\quad u_ y(0,y)=h(y), \] in an unbounded region contained in the ...
Kreith, Kurt +2 more
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Multiscale homogenization of nonlinear hyperbolic-parabolic equations
Applications of Mathematics, 2022zbMATH Open Web Interface contents unavailable due to conflicting licenses.
Dehamnia, Abdelhakim, Haddadou, Hamid
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Nonlinear hyperbolic volterra integrodifferential equations
Nonlinear Analysis: Theory, Methods & Applications, 1996The well posedness of the abstract Cauchy problem \[ u'(t) = Au(t) + \int^t_{t_0} K \bigl( t,s,u(s) \bigr) ds + f(t), \quad u(t_0) = u_0 \] is studied, \(A\) denoting a linear Hille-Yosida operator in the Banach space \((X,II \cdot II)\). The paper consists of different Sections, and includes the proof of various theorems. The last Section refers to an
Nagel, Rainer, Sinestrari, Eugenio
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On a Nonlinear Hyperbolic Volterra Equation
SIAM Journal on Mathematical Analysis, 1980We study questions of existence, boundedness and asymptotic behavior of the solutions of the initial value problem \[(*)\qquad \begin{array}{*{20}c} {u_t (t,x) - \int_0^t {a (t - s)\sigma (u_x (s,x))_x = f(t,x),\quad 0 < t < \infty ,\quad x \in R.} } \\ {u(0,x) = u_0 (x),\quad x \in R.} \\ \end{array} \] Here $a:R^ + = [0,\infty ) \to R,\sigma :R \to R,
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A Perturbation Method for Hyperbolic Equations with Small Nonlinearities
SIAM Journal on Applied Mathematics, 1972A method of multiple scales is developed for the generation of uniformly valid asymptotic solutions of initial value problems for nonlinear wave equations. The method is applicable when the nonlinearities are small and only involve the first derivatives of the dependent variable.
Chikwendu, S. C., Kevorkian, J.
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On Nonlinear Hyperbolic Functional Differential Equations
Mathematische Nachrichten, 2000The author proves existence of weak solutions of certain second-order evolution equations. The results are applied to higher-order nonlinear hyperbolic functional-differential equations.
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Symmetric hyperbolic equations in the nonlinear elasticity theory
Computational Mathematics and Mathematical Physics, 2008Summary: Concerning the formulation of nonlinear elasticity equations in the form of symmetric hyperbolic systems, the article surveys basic results of long-time studies performed under the direction of the first author. The underlying principles developed therein are stated, and some inaccuracies and errors are corrected.
Godunov, S. K., Peshkov, I. M.
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Second Order Hyperbolic Equations with Small Nonlinearities
SIAM Journal on Applied Mathematics, 1978A second order partial differential equation which describes the propagation of one-dimensional nonlinear waves in a bounded, inhomogeneous, dissipative medium is analyzed using the method of multiple scales. The conditions under which the oppositely traveling components of the nonlinear motion uncouple to first order are given.
Seymour, Brian R., Mortell, Michael P.
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Hyperbolicity of the Nonlinear Models of Maxwell?s Equations
Archive for Rational Mechanics and Analysis, 2004The class of nonlinear models of electromagnetism is considered [see \textit{B. D. Coleman} and \textit{E. H. Dill}, Z. Angew. Math. Phys. 22, 691--702 (1971; Zbl 0218.35072)]. To describe the electromagnetic field \((B,D)\) its energy density \(W(B,D)\) is used. The models are constructed on basis of conservation laws.
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