Smoothing estimates for nondispersive equations
 728 Downloads
 1 Citations
Abstract
1 Introduction
Among them, a comprehensive analysis is presented in our previous paper [30] for global smoothing estimates by using two useful methods, that is, canonical transformations and the the comparison principle. Canonical transformations are a tool to transform one equation to another at the estimate level, and the comparison principle is a tool to relate differential estimates for solutions to different equations. These two methods work very effectively under the dispersiveness conditions to induce a number of new or refined global smoothing estimates, as well as many equivalences between them. Using these methods, the proofs of smoothing estimates are also considerably simplified.
The objective of this paper is to continue the investigation by the same approach in the case when the dispersiveness breaks down. We will conjecture what we may call an ‘invariant estimate’ extending the smoothing estimates to the nondispersive case. Such an estimate yields the known smoothing estimates in dispersive cases, it is invariant under canonical transforms of the problem, and we will show its validity for a number of nondispersive evolution equations of several different types.

in the dispersive case it is equivalent to the usual estimate (1.2);

it does continue to hold for a variety of nondispersive equations, where \(\nabla a(\xi )\) may become zero on some set and when (1.2) fails;

it does take into account possible zeros of the gradient \(\nabla a(\xi )\) in the nondispersive case, which is also responsible for the interface between dispersive and nondispersive zone (e.g. how quickly the gradient vanishes);

it is invariant under canonical transformations of the equation.

in radially symmetric cases, we can use the comparison principle of radially symmetric type (Theorem 3.1);

in polynomial cases we can use the comparison principle of one dimensional type (Theorem 3.2);

in the homogeneous case with some information on the Hessian, we can use canonical transformation to reduce the general case to some wellknown model situations (Theorem 3.3);

around nondispersive points where the Hessian is nondegenerate, we can microlocalise and apply the canonical transformation based on the Morse lemma (Theorem 3.4);
In addition, we will derive estimates for equations with time dependent coefficients. In general, the dispersive estimates for equations with time dependent coefficients may be a delicate problem, with decay rates heavily depending on the oscillation in coefficients (for a survey of different results for the wave equation with lower order terms see, e.g. Reissig [26]; for more general equations and systems and the geometric analysis of the timedecay rate of their solution see [31] or [8]). However, we will show in Sect. 4 that the smoothing estimates still remain valid if we introduce an appropriate factor into the estimate. Such estimates become a natural extension of the invariant estimates to the time dependent setting.
We will explain the organisation of this paper. In Sect. 2, we list typical global smoothing estimates for dispersive equations, and then discuss their invariant form which we expect to remain true also in nondispersive situations. In Sect. 3, we establish invariant estimates for several types of nondispersive equations. The case of timedependent coefficients will be treated in Sect. 4. In Appendix A, we review our fundamental tools, that is, the canonical transformation and the comparison principle, which is used for the analysis in Sect. 3.
Finally we comment on the notation used in this paper. When we need to specify the entries of the vectors \(x,\xi \in {\mathbb R}^n\), we write \(x=(x_1,x_2,\ldots ,x_n)\), \(\xi =(\xi _1,\xi _2,\ldots ,\xi _n)\) without any notification. As usual we will denote \(\nabla =(\partial _1,\ldots ,\partial _n)\) where \(\partial _j=\partial _{x_j}\), \(D_x=(D_1,D_2\ldots ,D_n)\) where \(D_{j}=\sqrt{1}\,\partial _{j}\) (\(j=1,2,\ldots ,n\)), and view operators \(a(D_x)\) as Fourier multipliers. We denote the set of the positive real numbers \((0,\infty )\) by \({\mathbb R}_+\). Constants denoted by letter C in estimates are always positive and may differ on different occasions, but will still be denoted by the same letter.
2 Invariant smoothing estimates for dispersive equations
2.1 Smoothing estimates for dispersive equations
Theorem 2.1
 Suppose \(n\ge 1\), \(m>0\), and \(s>1/2\). Then we have$$\begin{aligned} {\left\ {{\left\langle {x}\right\rangle }^{s}D_x^{(m1)/2}e^{ita(D_x)}\varphi (x)}\right\ }_{L^2{\left( {{\mathbb R}_t\times {\mathbb R}^n_x}\right) }} \le C{\left\ {\varphi }\right\ }_{L^2{\left( {{\mathbb R}^n_x}\right) }}. \end{aligned}$$(2.1)
 Suppose \(m>0\) and \((mn+1)/2<\alpha <(m1)/2\). Or, in the elliptic case \(a(\xi )\ne 0\) \((\xi \ne 0)\), suppose \(m>0\) and \((mn)/2<\alpha <(m1)/2\). Then we have$$\begin{aligned} {\left\ {{\left {x}\right }^{\alpha m/2}D_x^{\alpha }e^{ita(D_x)}\varphi (x)}\right\ }_ {L^2{\left( {{\mathbb R}_t\times {\mathbb R}^n_x}\right) }} \le C{\left\ {\varphi }\right\ }_{L^2{\left( {{\mathbb R}^n_x}\right) }}. \end{aligned}$$(2.2)
 Suppose \(n1>m>1\), but in the elliptic case \(a(\xi )\ne 0\) \((\xi \ne 0)\) suppose \(n>m>1\). Then we have$$\begin{aligned} {\left\ {{\left\langle {x}\right\rangle }^{m/2}{\left\langle {D_x}\right\rangle }^{(m1)/2}e^{ita(D_x)}\varphi (x)}\right\ }_ {L^2{\left( {{\mathbb R}_t\times {\mathbb R}^n_x}\right) }} \le C{\left\ {\varphi }\right\ }_{L^2{\left( {{\mathbb R}^n_x}\right) }}. \end{aligned}$$(2.3)
In the Schrödinger equation case \(a(\xi )=\xi ^2\) and for \(n\ge 3\), estimate (2.1) was obtained by BenArtzi and Klainerman [2]. It follows also from a sharp local smoothing estimate by Kenig et al. [15, Theorem 4.1]), and also from the one by Chihara [5] who treated the case \(m>1\). For the range \(m>0\) and any \(n\ge 1\) it was obtained in [30, Theorem 5.1].
Compared to (2.1), the estimate (2.2) is scaling invariant with the homogeneous weights \(x^{s}\) instead of nonhomogenous ones \({\left\langle {x}\right\rangle }^{s}\). The estimate (2.2) was obtained in [30, Theorem 5.2], and it is a generalisation of the result by Kato and Yajima [14] who treated the case \(a(\xi )=\xi ^2\) with \(n\ge 3\) and \(0\le \alpha <1/2\), or with \(n=2\) and \(0<\alpha <1/2\), and also of the one by Sugimoto [35] who treated elliptic \(a(\xi )\) of order \(m=2\) with \(n\ge 2\) and \(1n/2<\alpha <1/2\).
The smoothing estimate (2.3) is of yet another type replacing \(D_x^{(m1)/2}\) by its nonhomogeneous version \({\left\langle {D_x}\right\rangle }^{(m1)/2}\), obtained in [30, Corollary 5.3]. It is a direct consequence of (2.1) with \(s=m/2\) and (2.2) with \(\alpha =0\) (note also the \(L^2\)boundedness of \({\left\langle {D_x}\right\rangle }^{(m1)/2}(1+D_x^{(m1)/2})^{1}\)), and it also extends the result by Kato and Yajima [14] who treated the case \(a(\xi )=\xi ^2\) and \(n\ge 3\), the one by Walther [40] who treated the case \(a(\xi )=\xi ^m\), and the one by the authors [27] who treated the elliptic case with \(m=2\).
Theorem 2.2
 Suppose \(n\ge 1\), \(m>0\), and \(s>1/2\). Then we have$$\begin{aligned} {\left\ {{\left\langle {x}\right\rangle }^{s}{\left\langle {D_x}\right\rangle }^{(m1)/2}e^{ita(D_x)}\varphi (x)}\right\ } _{L^2{\left( {{\mathbb R}_t\times {\mathbb R}^n_x}\right) }} \le C{\left\ {\varphi }\right\ }_{L^2{\left( {{\mathbb R}^n_x}\right) }}. \end{aligned}$$(2.4)
 Suppose \(n\ge 1\), \(m\ge 1\) and \(s>1/2\). Then we have$$\begin{aligned} {\left\ {{\left\langle {x}\right\rangle }^{s}D_x^{(m1)/2}e^{ita(D_x)}\varphi (x)}\right\ }_{L^2{\left( {{\mathbb R}_t\times {\mathbb R}^n_x}\right) }} \le C{\left\ {\varphi }\right\ }_{L^2{\left( {{\mathbb R}^n_x}\right) }}. \end{aligned}$$(2.5)
The estimate (2.4) was established in [30, Theorem 5.4]. Consequently, (2.5) is a straightforward consequence of (2.4) and the \(L^2\)boundedness of the Fourier multiplier \(D_x^{(m1)/2}{\left\langle {D_x}\right\rangle }^{(m1)/2}\) with \(m\ge 1\). It is an analogue of (2.1) for operators \(a(D_x)\) with lower order terms, and also a generalisation of [15, Theorem 4.1] who treated essentially polynomial symbols \(a(\xi )\). For (2.5) in its full generality we refer to [30, Corollary 5.5].
Theorem 2.3
2.2 Invariant estimates
In terms of invariant estimates, we can also give another explanation to the reason why we do not have timeglobal estimate in Theorem 2.3. The problem is that the symbol of the smoothing operator \({\left\langle {D_x}\right\rangle }^{(m1)/2}\) does not vanish where the symbol of \(\nabla a(D_x)\) vanishes, as should be anticipated by the invariant estimate (2.7). If zeros of \(\nabla a(D_x)\) are not taken into account, the weight should change to the one as in estimate (2.3).
3 Smoothing estimates for nondispersive equations
3.1 Radially symmetric case
The following result states that we still have estimate (2.7) of invariant form suggested in Sect. 2.2 even for nondispersive equations in a general setting of the radially symmetric case. Remarkably enough, it is a straight forward consequence of the second comparison method of Corollary 5.8, and in this sense, it is just an equivalent expression of the translation invariance of the Lebesgue measure (see Appendix A.3):
Theorem 3.1
Proof
Noticing \(\nabla a(\xi )=f'(\xi )\), use Corollary 5.8 for \(\sigma (\rho )=f'(\rho )^{1/2}\) in each interval where f is strictly monotone. \(\square \)
Example 3.1
3.2 Polynomial case
Another remarkable fact is that we can obtain invariant estimate (2.7) for all differential equations with real constant coefficients if we use second comparison method of Corollary 5.9 (hence again it is just an equivalent expression of the translation invariance of the Lebesgue measure):
Theorem 3.2
Proof
Example 3.2
Some of normal forms listed listed in (1.4) in Introduction satisfy dispersiveness assumptions. Indeed, \(a(\xi _1,\xi _2)=\xi _1^3+\xi _2^3\) and \(\xi _1^3\xi _1\xi _2^2\) are homogeneous and satisfy the assumption (H). The other normal forms however satisfy neither (H) nor (L). For example, \(a(\xi _1,\xi _2)=\xi _1^3\) and \(\xi _1\xi _2^2\) are homogeneous but \(\nabla a(\xi _1,\xi _2)=0\) when \(\xi _1=0\) and \(\xi _2=0\) respectively. On the other hand \(a(\xi _1,\xi _2)=\xi _1^3+\xi _2^2\) and \(\xi _1\xi _2^2+\xi _1^2\) are not homogeneous and satisfy \(\nabla a(\xi _1,\xi _2)=0\) at the origin. See Example 3.5 for others. But even for them we still have invariant smoothing estimate (2.7) with \(s>1/2\) by Theorem 3.2.
3.3 Hessian at nondispersive points
Now we will present another approach to treat nondispersive equations. Recall that, in [30, Section 5], the method of canonical transformation effectively works to reduce smoothing estimates for dispersive equations listed in Sect. 2.1 to some model estimates. For example, as mentioned in the beginning of Appendix A.3, estimate (2.1) in Theorem 2.1 is reduced to model estimates (5.6) and (5.7) in Corollary 5.7. We explain here that this strategy works for nondispersive cases as well.
Lemma 3.1
Let \(a=\sigma \circ \psi \), with \(\psi :U\rightarrow {\mathbb R}^n\) satisfying \(\det D\psi (\xi )\not =0\) on an open set \(U\subset {\mathbb R}^n\). Then, for each \(\xi \in U\), \(\nabla a(\xi )=0\) if and only if \(\nabla \sigma (\psi (\xi ))=0\). Furthermore the ranks of \(\nabla ^2 a(\xi )\) and \(\nabla ^2\sigma (\psi (\xi ))\) are equal on \(\Gamma \) whenever \(\xi \in U\) and \(\nabla a(\xi )=0\).
Proof
On account of the above observations, we have the following result which states that invariant estimates (2.7), (2.8) and (2.9) with \(m=2\) still hold for a class of nondispersive equations:
Theorem 3.3
 Suppose \(n\ge 2\) and \(s>1/2\). Then we have$$\begin{aligned} {\left\ {{\left\langle {x}\right\rangle }^{s}\nabla a(D_x)^{1/2} e^{it a(D_x)}\varphi (x)}\right\ }_{L^2{\left( {{\mathbb R}_t\times {\mathbb R}^n_x}\right) }} \le C{\left\ {\varphi }\right\ }_{L^2{\left( {{\mathbb R}^n_x}\right) }}. \end{aligned}$$
 Suppose \((4n)/2<\alpha <1/2\), or \((3n)/2<\alpha <1/2\) in the elliptic case \(a(\xi )\ne 0\) (\(\xi \ne 0\)). Then we have$$\begin{aligned} {\left\ {{\left {x}\right }^{\alpha 1}\nabla a(D_x)^{\alpha }e^{ita(D_x)}\varphi (x)}\right\ }_ {L^2{\left( {{\mathbb R}_t\times {\mathbb R}^n_x}\right) }} \le C{\left\ {\varphi }\right\ }_{L^2{\left( {{\mathbb R}^n_x}\right) }}. \end{aligned}$$
 Suppose \(n>4\), or \(n>3\) in the elliptic case \(a(\xi )\ne 0\) \((\xi \ne 0).\) Then we have$$\begin{aligned} {\left\ {{\left\langle {x}\right\rangle }^{1}{\left\langle {\nabla a(D_x)}\right\rangle }^{1/2}e^{ita(D_x)}\varphi (x)}\right\ }_ {L^2{\left( {{\mathbb R}_t\times {\mathbb R}^n_x}\right) }} \le C{\left\ {\varphi }\right\ }_{L^2{\left( {{\mathbb R}^n_x}\right) }}. \end{aligned}$$
Proof
Example 3.3
3.4 Isolated critical points
Next we consider more general operators \(a(\xi )\) of order m which may have some lower order terms. Then even the most favourable case \(\det \nabla ^2a(\xi )\not =0\) does not imply the dispersive assumption \(\nabla a(\xi )\not =0\). The method of canonical transformation, however, can also allow us to treat this problem by obtaining localised estimates near points \(\xi \) where \(\nabla a(\xi )=0\).
Theorem 3.4
 Suppose \(n\ge 1\), \(m\ge 1\), and \(s>1/2\). Then we have$$\begin{aligned} {\left\ {{\left\langle {x}\right\rangle }^{s}\nabla a(D_x)^{1/2} e^{it a(D_x)}\varphi (x)}\right\ }_{L^2{\left( {{\mathbb R}_t\times {\mathbb R}^n_x}\right) }} \le C{\left\ {\varphi }\right\ }_{L^2{\left( {{\mathbb R}^n_x}\right) }}. \end{aligned}$$
 Suppose \(n>2\) and \(m\ge 1\). Then we have$$\begin{aligned} {\left\ {{\left\langle {x}\right\rangle }^{1}{\left\langle {\nabla a(D_x)}\right\rangle }^{1/2} e^{it a(D_x)}\varphi (x)}\right\ }_{L^2{\left( {{\mathbb R}_t\times {\mathbb R}^n_x}\right) }} \le C{\left\ {\varphi }\right\ }_{L^2{\left( {{\mathbb R}^n_x}\right) }}. \end{aligned}$$
Example 3.4
Example 3.5
4 Equations with timedependent coefficients
Theorem 4.1
Let \([\alpha ,\beta ]\subset [\infty ,+\infty ]\). Assume that function \(c=c(t)\) is continuous on \([\alpha ,\beta ]\) and that \(c\not =0\) on \((\alpha ,\beta )\). Let \(u=u(t,x)\) be the solution of Eq. (4.1) with \(b(t,\xi )=c(t)a(\xi )\), where \(a=a(\xi )\) satisfies assumptions of any part of Theorems 2.1, 2.2, 2.3, 3.1, 3.3 or 3.4. Then the smoothing estimate of the corresponding theorem holds provided we replace \(L^2({\mathbb R}_t,{{\mathbb R}^n_x})\) by \(L^2([\alpha ,\beta ],{{\mathbb R}^n_x})\), and insert \(c(t)^{1/2}\) in the left hand side norms.
We note that it is possible that \(\alpha =\infty \) and that \(\beta =+\infty \), in which case by continuity of c at such points we simply mean that the limits of c(t) exist as \(t\rightarrow \alpha +\) and as \(t\rightarrow \beta \).
References
 1.BenArtzi, M., Devinatz, A.: Local smoothing and convergence properties of Schrödinger type equations. J. Funct. Anal. 101, 231254 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
 2.BenArtzi, M., Klainerman, S.: Decay and regularity for the Schrödinger equation. J. Anal. Math. 58, 2537 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
 3.BenArtzi, M., Koch, H., Saut, J.C.: Dispersion estimates for third order equations in two dimensions. Commun. Partial Differ. Equ. 28, 19431974 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
 4.BenArtzi, M., Nemirovsky, J.: Remarks on relativistic Schrödinger operators and their extensions. Ann. Inst. H. Poincare Phys. Theor. 67, 2939 (1997)MathSciNetzbMATHGoogle Scholar
 5.Chihara, H.: Smoothing effects of dispersive pseudodifferential equations. Commun. Partial Differ. Equ. 27, 19532005 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
 6.Chihara, H.: Resolvent estimates related with a class of dispersive equations. J. Fourier Anal. Appl. 14, 301325 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
 7.Constantin, P., Saut, J.C.: Local smoothing properties of dispersive equations. J. Am. Math. Soc. 1, 413439 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
 8.CruzUribe, D., Fiorenza, A., Ruzhansky, M., Wirth, J.: Variable Lebesgue spaces and hyperbolic systems. In: Advanced Courses in Mathematics–CRM Barcelona, vol. 27. Birkhäuser (2014)Google Scholar
 9.Ghidaglia, J.M., Saut, J.C.: Nonelliptic Schrödinger equations. J. Nonlinear Sci. 3, 169195 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
 10.Hoshiro, T.: Mourre’s method and smoothing properties of dispersive equations. Commun. Math. Phys. 202, 255265 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
 11.Hoshiro, T.: Decay and regularity for dispersive equations with constant coefficients. J. Anal. Math. 91, 211230 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
 12.Kato, T.: Wave operators and similarity for some nonselfadjoint operators. Math. Ann. 162, 258279 (1966)MathSciNetCrossRefzbMATHGoogle Scholar
 13.Kato, T.: On the Cauchy problem for the (generalized) Kortewegde Vries equation. In: Studies in Applied Mathematics. Adv. Math. Suppl. Stud., vol. 8, pp. 93128. Academic Press, New York (1983)Google Scholar
 14.Kato, T., Yajima, K.: Some examples of smooth operators and the associated smoothing effect. Rev. Math. Phys. 1, 481496 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
 15.Kenig, C.E., Ponce, G., Vega, L.: Oscillatory integrals and regularity of dispersive equations. Indiana Univ. Math. J. 40, 3369 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
 16.Kenig, C.E., Ponce, G., Vega, L.: Wellposedness and scattering results for the generalized Kortewegde Vries equation via the contraction principle. Commun. Pure Appl. Math. 46, 527620 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
 17.Kenig, C.E., Ponce, G., Vega, L.: Small solutions to nonlinear Schrödinger equations. Ann. Inst. H. Poincaré Anal. Non Linéaire 10, 255288 (1993)MathSciNetzbMATHGoogle Scholar
 18.Kenig, C.E., Ponce, G., Vega, L.: On the generalized BenjaminOno equation. Trans. Am. Math. Soc. 342, 155172 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
 19.Kenig, C.E., Ponce, G., Vega, L.: On the Zakharov and ZakharovSchulman systems. J. Funct. Anal. 127, 204234 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
 20.Kenig, C.E., Ponce, G., Vega, L.: Smoothing effects and local existence theory for the generalized nonlinear Schrodinger equations. Invent. Math. 134, 489545 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
 21.Linares, F., Ponce, G.: On the DaveyStewartson systems. Ann. Inst. H. Poincaré Anal. Non Linéaire 10, 523548 (1993)MathSciNetzbMATHGoogle Scholar
 22.Manganaro, N., Parker, D.F.: Similarity reductions for variablecoefficient coupled nonlinear Schrodinger equations. J. Phys. A Math. Gen. 26, 40934106 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
 23.Morii, K.: Timeglobal smoothing estimates for a class of dispersive equations with constant coefficients. Ark. Mat. 46, 363375 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
 24.Nakkeeran, K.: Optical solitons in erbium doped fibers with higher order effects. Phys. Lett. A 275, 415418 (2000)CrossRefzbMATHGoogle Scholar
 25.Pelinovsky, D., Yang, J.: Instabilities of multihamp vector solitons in coupled nonlinear Schrödinger euqations. Stud. Appl. Math. 115, 109137 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
 26.Reissig, M.: $L_pL_q$ decay estimates for wave equations with timedependent coefficients. J. Nonlinear Math. Phys. 11, 534548 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
 27.Ruzhansky, M., Sugimoto, M.: Global $L^2$boundedness theorems for a class of Fourier integral operators. Commun. Partial Differ. Equ. 31, 547569 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
 28.Ruzhansky, M., Sugimoto, M.: A smoothing property of Schrödinger equations in the critical case. Math. Ann. 335, 645673 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
 29.Ruzhansky, M., Sugimoto, M.: Structural resolvent estimates and derivative nonlinear Schrödinger equations. Commun. Math. Phys. 314, 281304 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
 30.Ruzhansky, M., Sugimoto, M.: Smoothing properties of evolution equations via canonical transforms and comparison principle. Proc. Lond. Math. Soc. 105, 393423 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
 31.Ruzhansky, M., Wirth, J.: Dispersive estimates for hyperbolic systems with timedependent coefficients. J. Differ. Equ. 251, 941969 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
 32.Shrira, V.I.: On the propagation of a threedimensional packet of weakly nonlinear internal gravity waves. Int. J. Non Linear Mech. 16, 129138 (1981)MathSciNetCrossRefzbMATHGoogle Scholar
 33.Simon, B.: Best constants in some operator smoothness estimates. J. Funct. Anal. 107, 6671 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
 34.Sjölin, P.: Regularity of solutions to the Schrödinger equation. Duke Math. J. 55, 699715 (1987)MathSciNetCrossRefzbMATHGoogle Scholar
 35.Sugimoto, M.: Global smoothing properties of generalized Schrödinger equations. J. Anal. Math. 76, 191204 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
 36.Sugimoto, M.: A Smoothing property of Schrödinger equations along the sphere. J. Anal. Math. 89, 1530 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
 37.Tan, B., Boyd, J.: Coupledmode envelope solitary waves in a pair of cubic Schrödinger equations with cross modulation: analytical solution and collisions. Chaos Solitons Fractals (1999)Google Scholar
 38.Vega, L.: Schrödinger equations: Pointwise convergence to the initial data. Proc. Am. Math. Soc. 102, 874878 (1988)MathSciNetzbMATHGoogle Scholar
 39.Walther, B.G.: A sharp weighted $L^2$estimate for the solution to the timedependent Schrödinger equation. Ark. Mat. 37, 381393 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
 40.Walther, B.G.: Regularity, decay, and best constants for dispersive equations. J. Funct. Anal. 189, 325335 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
 41.Watanabe, K.: Smooth perturbations of the selfadjoint operator $\vert \Delta \vert ^{\alpha /2}$. Tokyo J. Math. 14, 239250 (1991)MathSciNetCrossRefGoogle Scholar
 42.Zen, F.P., Elim H.I.: Multisoliton solution of the integrable coupled nonlinear Scrödinger equation of Manakov type. arXiv:solvint/9901010Google Scholar
Copyright information
Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.