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Exact solutions of semilinear radial Schrodinger equations by separation of group foliation variables

Explicit solutions are obtained for a class of semilinear radial Schrodinger equations with power nonlinearities in multi-dimensions. These solutions include new similarity solutions and other new group-invariant solutions, as well as new solutions that are not invariant under any symmetries of this class of equations. Many of the solutions have interesting analytical behavior connected with blow-up and dispersion. Several interesting nonlinearity powers arise in these solutions, including the case of the critical (pseudo-conformal) power. In contrast, standard symmetry reduction methods lead to nonlinear ODEs for which few if any explicit solutions can be derived by standard integration methods.


Introduction
An interesting class of nonlinear wave equations consists of the semilinear Schrödinger equations ıu t = u rr + mu r /r + k|u| p u, p = 0, k = 0 (1.1) for u(t, r) ∈ C, where p ∈ R is a nonlinearity power, k ∈ R is the nonlinearity coefficient, and m ∈ R is a spatial-derivative coefficient. When m is a positive integer, this wave equation (1.1) physically describes a general model for the slow modulation of radial waves in a weakly nonlinear, dispersive, isotropic medium [1] in m + 1 dimensions, with radial coordinate r. When m is zero, the equation similarly is a model for slow modulation of waves in a onedimensional, weakly nonlinear, dispersive medium, where r is the full-line coordinate. In all other cases equation (1.1) can be interpreted instead as modelling the slow modulation of two-dimensional radial waves in a planar, weakly nonlinear, dispersive medium containing a point-source disturbance at the origin, represented by an extra modulation term (m − 1)u r /r [5]. This interpretation can be applied more generally for m = 2. Hereafter we will call (1.1) the radial NLS equation and write m = n − 1 (1.2) without any restriction on n ∈ R. Exact solutions have an important role in the study of the radial NLS equation (1.2), particularly for understanding blow-up, dispersive behaviour, attractors, and critical dynamics, as well as for testing numerical solution methods. Stability and global behaviour of solutions to the initial-value problem depend on [2,1] the effective dimension n = m + 1, the nonlinearity power p, and the sign of the interaction coefficient k. Specifically, for p ≥ 4/n > 0 and k > 0, some solutions exhibit a finite time blow-up such that |u(t, r)| → ∞ as t → T < ∞.
In the case p = 4/n > 0, which is known as the critical power, a special class of blow-up solutions is rigorously known to have the form [1] u(t, r) = (T − t) −n/2 U(ξ) exp(ı(ω + r 2 /4)/(T − t)), ξ = r/(T − t), ω = 0, (1.3) which is invariant under a certain pseudo-conformal subgroup in the full symmetry group of equation (1.1), where U(ξ) satisfies a complex nonlinear 2nd order ODE U ′′ + (n − 1)ξ −1 U ′ + ωU + k|U| 4/n U = 0. (1.4) In the supercritical case p > 4/n > 0, a general class of blow-up solutions is believed [1] to asymptotically approach an exact similarity form Both ODEs (1.4) and (1.6) are, however, intractable to solve by standard ODE integration techniques [3,4] such as symmetry reduction and integrating factors. In fact, as summarized in recent work [5], the only explicit solutions which are known to-date for n = 1 (m = 0) consist of the obvious constant solution U = (−ω/k) n/4 exp(ıφ) for the ODE (1.4). In this paper we will obtain new explicit exact solutions to the radial NLS equation (1.1) for n = 1 (m = 0) by applying a symmetry group method which has been used successfully in previous work [6,7,8] to find explicit blow-up and dispersive solutions to semilinear radial wave equations and semilinear radial heat equations with power nonlinearities in multi-dimensions. The method uses the group foliation equations associated with onedimensional subgroups of the point symmetry group of a given nonlinear PDE [9]. These equations consist of an equivalent first-order PDE system whose independent and dependent variables are respectively defined by the invariants and differential invariants of a given point symmetry subgroup. Each solution of the system geometrically corresponds to an explicit one-parameter family of exact solutions of the original nonlinear PDE, such that the family is closed under the given point symmetry subgroup acting in the solution space of the PDE. In the case of a PDE with power nonlinearities, the form of the resulting group-foliation system allows explicit solutions to be found by a systematic separation technique in terms of the group-invariant variables. We will use an improved version of this technique, which is able to yield a much wider set of solutions.
Our results include explicit blow-up solutions having the group-invariant forms (1.3) and (1.5), plus explicit blow-up solutions with a non-invariant form, in addition to explicit dispersive solutions, explicit standing wave solutions, and explicit monopole solutions. Among the new solutions, some are found to hold only for non-integer values of n − 1 (= m), which we interpret as radial planar solutions in the presence of a modulation point-source at the origin.
In Sec. 2, the group-foliation method is set up based on the symmetry group of the radial NLS equation (1.1). The improved separation technique used for finding explicit solutions of the group foliation equations is then applied in Sec. 3, and the resulting exact solutions of the radial NLS equation along with their basic analytical features are summarized in Sec. 4. Finally, some concluding remarks are made in Sec. 5.

Symmetries and group foliations
The group of point symmetries of the radial NLS equation (1.1) for n = 1 (m = 0) is wellknown [10,11] to be generated by the following point transformations acting on (t, r, u,ū): phase rotation X phas. = ıu∂/∂u − ıū∂/∂ū for all p, The full group of point symmetries is obtained by compositions of these transformations (2.5)-(2.8).
A group foliation can be constructed using any linear combination X of symmetry generators (2.1)-(2.4) such that X has a regular projectable [4] action on (t, r). In particular, X = c 1 X phas. + c 2 X trans. + c 3 X scal. + c 4 X inver. (2.9) projects to (c 2 + 2c 3 t + c 4 t 2 )∂/∂t + (c 3 r + c 4 tr)∂/∂r, whose action on (t, r) is regular if and only if c 2 2 + c 2 3 + c 2 4 = 0 (2.10) (with c 4 = 0 if p = 4/n). For any symmetry generator X of the form (2.9)-(2.10), a group foliation consists of converting the radial NLS equation (1.1) into a system of firstorder equations, called the group resolving system, for the orbits of the one-dimensional symmetry group generated by X. This system is naturally formulated in terms of a complete set of invariants x(t, r), v(t, r, u),v(t, r,ū) and a complete set of 1st-order differential invariants G(t, r, u, u t , u r ),Ḡ(t, r,ū,ū t ,ū r ), H(t, r, u, u t , u r ),H(t, r,ū,ū t ,ū r ) of X, which always can be chosen so that the phase-rotation symmetry (2.1) leaves x invariant and acts equivariantly on v,v, G, H,Ḡ,H. As a consequence, the solution space {u = f (t, r)} of the radial NLS equation (1.1) can be recovered from the phase-equivariant solution space )} of the group-resolving system by integration of the 1st order complex differential equations for u(t, r) G(t, r, u, u t , u r ) = g(x(r, t), v(t, r, u),v(t, r,ū)), H(t, r, u, u t , u r ) = h(x(r, t), v(t, r, u),v(t, r,ū)), (2.11) where this pair of differential equations can be reduced to two quadratures due to their builtin invariance with respect to the two-dimensional symmetry group generated by X and X phas. . These two quadratures thereby produce a two-parameter family of radial NLS solutions u = f (t, r, c 1 , c 2 ) from each phase-equivariant solution (G = g(x, v,v), H = h(x, v,v)) of the group-resolving system. Note that the invariance of the differential equations (2.11) under phase-rotations is essential for having a sufficiently large symmetry group to allow integrating them to quadratures. We now set up the group-resolving systems for each of the symmetry generators given by time-translation (2.2), scaling (2.3), and inversion (2.4). A general remark is that groupresolving systems arising from different choices of symmetry groups are not related to each other by any simple transformation, while the form of any specific group-resolving system depends on the complexity of the expressions for the symmetry generator X and for the invariants x, v,v and differential invariants G,Ḡ, H,H. Accordingly, we will not consider group-resolving systems given by linear combinations of the generators (2.1)-(2.4), such as an optimal set with respect to conjugacy in the full symmetry group, since such systems have a more complicated form that makes it harder to find explicit solutions.
2.1. Time-translation-group resolving system. To proceed, we first write down the obvious invariants Similarly, we write down the obvious differential invariants trans. H = 0 and X phas. G = ıG, X phas. H = ıH, where X trans. is the first-order prolongation of the time-translation generator (2.2) and X (1) phas. is the first-order prolongation of the phase-rotation generator (2.1). Here x, v andv are mutually independent, while G and H are related by equality of mixed r, t derivatives on u t and u r , which gives where D r , D t denote total derivatives with respect to r, t. Furthermore, v,v, G, H are related through the radial NLS equation (1.1) by into equations (2.14)-(2.15) and use equation (2.12) combined with the chain rule to arrive at a first-order system with independent variables x, v,v, and dependent variables G, H (and their complex conjugates). These equations will be called the time-translation-group resolving system for the radial NLS equation (1.1).
The respective solution spaces of equation (1.1) and system (2.16) are related by a groupinvariant mapping that is defined through the invariants (2.12) and differential invariants (2.13), and that preserves phase-rotation symmetry. The proof of Lemma 1 will be given in Sec. 3

2.2.
Scaling-group resolving system. We proceed by writing down the invariants and differential invariants determined by the scaling generator (2.3) and its first-order prolongation. A simple choice of invariants is given by scal. H = 0 and X (1) phas. G = ıG, X Here the invariants x, v andv are again mutually independent, while the differential invariants G and H are related by equality of mixed r, t derivatives on u t and u r , which gives In addition, v,v, G, H are related through the radial NLS equation (1.1) by into equations (2.23)-(2.24) and apply the chain rule with equation (2.21) to get a first-order system with independent variables x, v,v, and dependent variables G, H (and their complex conjugates). These equations will be called the scaling-group resolving system for the radial NLS equation (1.1).
Similarly to the group foliation based on time-translation, here the respective solution spaces of equation (1.1) and system (2.25) are related by a group-invariant mapping, as defined through the invariants (2.21) and differential invariants (2.22), preserving phaserotation symmetry.

the constants of integration of the pair of parametric first-order ODEs
which are invariant under the scaling symmetry (2.3) and the phase rotation symmetry (2.1).
The proof of Lemma 3 is given in Sec. 3.2. Through the steps in this proof, a simple correspondence can be derived between similarity solutions of the radial NLS equation (1.1) and a particular class of solutions of the scaling-group resolving system (2.25) as follows.

Lemma 4. There is a one-to-one correspondence between two-parameter families of similarity solutions
of the radial NLS equation (1.1) and phase-equivariant solutions of the scaling-group resolving system (2.25) that satisfy the condition This correspondence establishes a relationship between classical similarity reduction of the radial NLS equation (1.1) under scaling symmetry and a reduction of the scaling-group resolving system (2.16) under condition (2.29).
2.3. Inversion-group resolving system. From the inversion generator (2.4), we first write down the mutually independent invariants Next we write down the simplest choice of mutually independent differential invariants G(t, r, u, u t , u r ) and H(t, r, u, u t , u r ) satisfying X (1) inver. G = X (1) inver. H = 0 and X (1) phas. G = ıG, X (1) phas. H = ıH: G = r 2+n/2 exp(ır 2 /(4t)) u t + ru r /t + (n/(2t) + ır 2 /(4t 2 ))u , (2.31a) These differential invariants are related by equality of mixed r, t derivatives on u t and u r , and by the radial NLS equation (1.1), which yields into equations (2.32)-(2.33) and applying the chain rule with equation (2.30), we get a first-order system with independent variables x, v,v, and dependent variables G, H (and their complex conjugates). These equations will be called the inversion-group resolving system for the radial NLS equation (1.1). The respective solution spaces of equation (1.1) and system (2.34) are related by a groupinvariant mapping that is defined through the invariants (2.30) and differential invariants (2.31) similarly to the group foliations based on time-translation and scaling, and that preserves phase-rotation symmetry.
A proof will be given in Sec. 3.3. Through the steps in the proof, there is a simple correspondence between pseudo-conformal solutions of the radial NLS equation (1.1) and a particular class of solutions of the inversion-group resolving system (2.25).

Lemma 6.
There is a one-to-one correspondence between two-parameter families of pseudoconformal solutions of the radial NLS equation (1.1) and phase-equivariant solutions of the inversion-group resolving system (2.34) that satisfy the condition This result gives a direct relationship between classical reduction of the radial NLS equation (1.1) under the pseudo-conformal symmetry group and a reduction of the inversion-group resolving system (2.16) under condition (2.38).

Solutions of the group-resolving systems
We will now explain how a group-invariant map relating solutions ) of a group-resolving system and two-parameter families of solutions u = f (t, r, c 1 ) exp(ıc 2 ) of the radial NLS equation (1.1) arises from integration of the pair of differential equations (2.11).
Let y be a canonical coordinate given by Xy = 1 where X is the symmetry generator used in constructing the group foliation. A change of variables in the differential equations (2.11) via the point transformation (t, r, u,ū) → (y, x, v,v) then yields v y =g(x, v,v) and v x =h(x, v,v), whereg andh are each given by a linear combination of G and H with coefficients depending on x, v,v. This pair of 1st order differential equations for v(y, x) inherits the invariance of the differential equations (2.11) with respect to the symmetry generators X and X phas. , so consequently,g andh can be restricted to have the phase- Hence the 1st order differential equations can be written as a pair of parametric ODEs exhibiting explicit symmetry invariance with respect to X = ∂/∂y and X phas. = ıv∂/∂v − ıv∂/∂v. It is straightforward to integrate these ODEs (3.1) after v = A exp(ıΦ) is expressed in polar form, giving
Note that these expressions (3.17)-(3.18) for (H, G) have the equivalent phase-equivariant form given by where ν, µ, ǫ + κ are certain functions of h 1 (x) and h 2 (x). The separation of variables ansatz (3.17)-(3.18) for (H, G) is more general than the twoterm ansatzes used in previous work [6,7,8] where the terms in G were restricted to contain the same powers as the terms in H, e.g.
Under the improved ansatz (3.17)-(3.18), a group-resolving system (3.14) will reduce to a single equation containing the monomial powers v, v a+1va , v 2a+1v2a , v 3a+1v3a , v 1+p/2vp/2 , v a+1+p/2va+p/2 , with coefficients that depend on the complex functions h 1 (x), h 2 (x), the exponents a, p, and the dimension n. From all possible balances among these monomial powers, five cases arise: a = −p/2; a = p/2; a = p/4; a = p/6; a = −p/2, p/2, p/4, p/6. In each case, the separate coefficients of the monomials must vanish, resulting in an overdetermined system of algebraic-differential equations for the unknowns Re h 1 (x), Im h 1 (x), Re h 2 (x), Im h 2 (x), a, p, n. Such systems can be solved by a systematic integrability analysis, which we have carried out using the computer algebra program Crack [12].
3.1. Results for the time-translation-group resolving system. The overdetermined systems of algebraic-differential equations that arise from reduction of the time-translationgroup resolving system (2.16) under the separation of variables ansatz (3.17)-(3.18) are found to admit non-zero solutions (h 1 (x), h 2 (x)) only in the cases a = p/2, a = p/4, and a = 1/n. For p = 0 and n = 1, the solutions are given by: (3.28)

29).) This 1st order ODE (3.31) is a Riccati equation which can be converted into Bessel's equation by the transformation
giving The form of solutions depends on the sign of C 1 : Hence yields the general solution for h 1 (x). Then ODE (3.30) for h 2 (x) becomes To integrate this equation (3.36), we consider the cases C 1 > 0 and C 1 < 0 separately. For the case C 1 < 0, we will use the Bessel function identity First apply this identity to the right-hand-side of equation (3.36) with z = |C 1 |x and µ = ν ∓ 1 = ∓n/2: Then equation (3.36) can be directly integrated to get Hence we obtain where the signs are determined from ±(1 − n/2) ≥ 0, and where f ν is given by the linear combination of Bessel functions (3.33). The case C 1 > 0 is similar but uses the modified Bessel function identity This leads to (3.48) , k > 0, n = 2, 3; (3.49) (3.50) For each phase-equivariant solution (G = g(x, |v|)v, H = h(x, |v|)v) of the timetranslation-group resolving system (2.16), the differential invariants (2.13) of X trans. yield a pair of parametric ODEs expressed in terms of the invariants x = r, v = u,v =ū and the canonical coordinate y = t of X trans. . These ODEs determine a two-parameter family of solutions u = f (t, r, c 1 ) exp(ıc 2 ) of the radial NLS equation (1.1), corresponding to orbits of the two-dimensional symmetry group generated by X trans. and X phas. . In the notation (3.1), withĝ = g andĥ = h, the explicit polar form u = A exp(ıΦ) of the solution families is given by the line integral formula Under the change of variables t = y, r = x, u = v, each such solution determines a function (3.9) from which the differential invariants (2.13) of X trans. are given by as written in terms of the functions g = (ln |v| + ı arg v) y and h = (ln |v| + ı arg v) x . These two functions are related by the differential identity and the radial NLS equation Now, from the relation |v| = |f (y + c 1 , x)|, since |f | c 1 = |f | y = 0 holds locally in the (y, x) plane, the implicit function theorem can be used to express y+c 1 = F (x, |v|) in terms of some function F . When this expression is substituted into g and h, they each become a function of just x and |v|. Hence the differential invariants (3.61) become phase-equivariant functions of x, v,v, which satisfy the time-translation-group resolving system (2.16) as consequence of equations (3.62)-(3.63). Finally, consider the case of solutions of the radial NLS equation (1.1) given by the twoparameter form u = f (t, r, c 1 ) exp(ıc 2 ) with |f | t = 0, (arg f ) t = const., and |f | c 1 = 0 holding in some open domain in the (t, r) plane. Each such solution determines a function (3.13) after the change of variables t = y, r = x, u = v. The differential invariants (2.13) of X trans. again have the form (3.61) in terms of two functions g = (ln |v| + ı arg v) y and h = (ln |v| + ı arg v) x , satisfying equations (3.62) and (3.63). Now, since |f | t = |f | y = 0 and |f | c 1 = 0 hold locally in the (y, x) plane, the implicit function theorem can be applied to the relation |v| = |f (y, x, c 1 )|, giving c 1 = F (x, |v|) in terms of some function F . This expression allows c 1 to be eliminated from h which then yields a function of just x and |v|, while g = (ln |f |+ı arg f ) y reduces to a constant as a consequence of |f | y = 0 and (arg f ) y = const.. The differential invariants (3.61) thereby become phase-equivariant functions of x, v,v, which satisfy the time-translation-group resolving system (2.16) due to equations (3.62)-(3.63).
This completes the proof of Lemma 1.
The invariants (2.30) and the canonical coordinate y = −1/t of X inver. can be used to write the differential invariants (2.31) of X inver. in the form of x, y-derivatives Then each solution (G = g(x, |v|)v, H = h(x, |v|)v) of the inversion-group resolving system (2.34) yields a pair of parametric ODEs (3.1) given by which determines a two-parameter family of solutions u = f (t, r, c 1 ) exp(ıc 2 ) of the radial NLS equation (1.1), corresponding to orbits of the two-dimensional symmetry group generated by X inver. and X phas. . These solution families are given by the polar form u = r −2/p A exp(ı(Φ − r 2 /(4t))) obtained from the line integral formula
Thirdly, the invariance property of each solution with respect to the symmetry group of the NLS equation will be listed.
A summary of these results is presented in Tables 1, 2, and 3.
The  Table 3.
Altogether, these 24 solutions encompass a wide range of different dynamical behaviours: static; time-periodic; dispersive; blow-up; and non-dispersive. In particular, one case of solution (4.3) exhibits a similarity blow-up (1.5) in which |u| → ∞ in a finite time t (though only for the subcritical power p = 2/n), and another case of this solution displays dispersion such that |u| → 0 for long times t → ∞ (again for the subcritical power p = 2/n). Other solutions exist for special nonlinearity powers p = 2/(2 − n), p = (2n − 6)/(2 − n) which are not distinguished by the symmetry structure of the radial NLS equation (1.1).
A detailed discussion of the interesting analytical features of all of the solutions will be given in a forthcoming paper [14].
The method we have used in the present work can be applied more generally to find explicit exact solutions to other complex (U(1)-invariant) semilinear evolutions in n ≥ 1 dimensions, such as derivative-type NLS equations ıu t = u xx + ı(a|u| p u x + b(|u| p ) x u) and mKdV-type equations u t = u xxx + a|u| p u x + b(|u| p ) x u in one dimension, and Landau-Ginzburg equations ıu t = △u + ıau + b|u| p u, Cahn-Hilliard equations u t = △(△u + au + b|u| p u), and Kuramoto-Sivashinsky equations u t = △ 2 u + a△u + b|∇u| p u in multi-dimensions.