In this paper, we consider the Schrodinger equation with quadratic potential i partial derivative/partial derivative t u = -Delta u + |x|(2) u in Rn+1, u(x,0) = f(x) is an element of L-2(R-n). Using Herrnite functions...
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In this paper, we consider the Schrodinger equation with quadratic potential i partial derivative/partial derivative t u = -Delta u + |x|(2) u in Rn+1, u(x,0) = f(x) is an element of L-2(R-n). Using Herrnite functions and some other classical tools, we give an elementary proof of the Kato-type smoothing estimate: for i not equal j not equal k, delta is an element of [0,1], and n >= 3, integral(2 pi)(0) integral(Rn) |u(x,t)|(2)/(x(i)(2) + x(j)(2) + x(k)(2))(delta) dxdt <= C parallel to f parallel to(2)(2). This is equivalent to proving a uniform L-2(R-n) boundedness result for a family of singularized Hermite projection kernels. As an application of the above estimate, we also prove the R-9 collapsing variable-type Strichartz estimate integral(2 pi)(0) integral(R3) |u(x,x,x,t)|(2) dxdt <= C parallel to(-Delta + |x|(2))f parallel to(2)(2) where x is an element of R-3.
In this paper, we investigate bounded traveling waves of the generalized nonlinear Klein-Gordon model equations by using bifurcation theory of planar dynamical systems to study the effects of horizontal singular strai...
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In this paper, we investigate bounded traveling waves of the generalized nonlinear Klein-Gordon model equations by using bifurcation theory of planar dynamical systems to study the effects of horizontal singular straight lines in nonlinear wave equations. Besides the well-known smooth traveling wave solutions and the non-smooth ones, four kinds of new bounded singular traveling wave solution are found for the first time. These singular traveling wave solutions are characterized by discontinuous second-order derivatives at some points, even though their first-order derivatives are continuous. Obviously, they are different from the singular traveling wave solutions such as compactons, cuspons, peakons. Their implicit expressions are also studied in this paper. These new interesting singular solutions, which are firstly founded, enrich the results on the traveling wave solutions of nonlinear equations. It is worth mentioning that the nonlinear equations with horizontal singular straight lines may have abundant and interesting new kinds of traveling wave solution.
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