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Encapsulated-vortex solutions to equivariant wave equations: Existence. (English) Zbl 0924.35143

The authors consider nonlinear Schrödinger and nonlinear Klein-Gordon equations \[ -Ju_t-\Delta u=\vec g(u) \tag{NLS} \] and \[ u_{tt}-\Delta u=\vec g(u),\tag{NLKG} \] where \(J\) is an invertible real skew-symmetric \(M\times M\) matrix and \(\vec g:\mathbb{R}^M\to \mathbb{R}^M\) is a continuous radial vector field such that \(\vec g(0)=0\) and \[ \vec g(y)= g\bigl(| y|\bigr) \widehat y\quad\text{ for }y\neq 0, \] where \(g: [0,\infty)\to \mathbb{R}\) is continuous with \(g(0)=0\) and \(\widehat y=y/| y|\cdot gt\) and \(u(x,t)= e^{vtK}v(x)\) is a solution of (NLS) on (NLKG), then \(v\) satisfies the elliptic equation \[ \Delta v+ \vec f(v)=0 \tag{1} \] where \(\vec f(y)=\vec g(y)+ \omega y\) with \(\omega=v\) for (NLS), \(w=v^2\) for (NLKG), \(v\) is a real constant and \(K\) is a real skew-symmetric \(M\times M\) matrix. The existence of twice differentiable solutions \(v\) of (1) such that \(v(x)\to 0\) as \(| x|\to\infty\) is established. Then \(u(x,t)= e^{vtK} v(x)\) is a localized standing wave solution of (NLS) on (NLKG). Under a velocity boost, such a standing wave becomes a spatially-localized travelling wave solution of the wave equation, that is, a multidimensional solitary wave.

MSC:

35Q53 KdV equations (Korteweg-de Vries equations)
35Q55 NLS equations (nonlinear Schrödinger equations)
35L70 Second-order nonlinear hyperbolic equations
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