Research Article | Open Access
Penghong Zhong, "Decay Rate and Energy Gap for the Singularity Solution of the Inhomogeneous Landau-Lifshitz Equation on ", Advances in Mathematical Physics, vol. 2018, Article ID 5370349, 15 pages, 2018. https://doi.org/10.1155/2018/5370349
Decay Rate and Energy Gap for the Singularity Solution of the Inhomogeneous Landau-Lifshitz Equation on
The singularity solution for the inhomogeneous Landau-Lifshitz (ILL) equation without damping term in -dimensional space was investigated. The implicit singularity solution was obtained for the case where the target space is on . This solution can be classified into four types that cover the global and local solutions. An estimation of the energy density of one of these types indicates its exact decay rate, which allows a global solution with finite initial energy under . Analysis of the four aperiodic solutions indicates that energy gaps that are first contributions to the literature of ILL will occur for particular coefficient settings, and these are shown graphically.
The nonlinear ferromagnetic chain model (FCM) has attracted the attention of physicists and mathematicians. For physicists, FCM represents the possibility of describing magnetic field density evolution with various external fields, while mathematicians study the various solutions or excitations in geometrically motive nonlinear models. Quantitative models depicting FCM were proposed by Landau and Lifshitz in 1935 , and the Landau-Lifshitz equation (LLE) was proposed to the dynamics of the nonequilibrium magnetism system: where is the variational sum of magnetic anisotropy energy and magnetic field energy, denotes magnetization intensity, is the cross product of Euclidean 3-space , and and are constants. If and , then LLE is the harmonic map heat flow (HMF). Otherwise, when , the Gilbert damping vanishes, and when and , the LLE degenerates into the Schrödinger map heat flow (SMF), which is an important equation of differential geometry. SMF can be regarded as a nonlinear Schrödinger equation that contains a derivative term. Although the existence, uniqueness, and the blow-up problem of some nonlinear Schrödinger equations [2–4] are clear, the theorem of SMF becomes more complicated and some further work still needs to be done. Similarly, comparing HMF and some general harmonic system (or even biharmonic equation) [5–8], the mapping system is more complicated than the nonmapping system due to the curvature flow of the Riemannian manifolds.
For clarity, we set and as the complex structure and metric on the Kähler manifold, respectively. Different manifolds and their covariant derivatives are and , respectively, and we define SMF as
Thus, (2) defines a mapping , which can degenerate into other subcases; for example, when and , (2) is the isotropic LLE without Gilbert damping, and if the manifold and complex structure are and (), respectively, then (2) is the hyperbolic isotropic LLE without Gilbert damping.
If and , then (2) can transform into a Schrödinger-type equation using the Hashimoto transform. Under dimensional cylindrical coordinates, let , and the curvature and torsion are, respectively, Then, employing the Hashimoto transform,and (2) transforms to
If , (5) is the standard cubic Schrödinger equation for which the Lax pair is obtained, and explicit solutions can be constructed using the Bäcklund transformation. Hence, (5) can be regarded as a simple case of (2). Similar to the inhomogeneous Schrödinger equation, (2) (or (1)) can be extended to the inhomogeneous expression. If we add the inhomogeneous term to (2), SME can be generalized into the inhomogeneous SME (ILL) proposed by Balakrishnan :where the inhomogeneous term, , is a scalar function. Under radially symmetrical coordinates, (6) can be expressed as
Daniel et al.  identified the integrable model (8) by analyzing the singularity structure of its solutions and also discussed Lax pairs, Bäcklund transformation, and soliton-like solutions of (7) (or (8)). They showed that inhomogeneous terms play an important role, and Painlevé analysis indicated that (7) (or (8)) was integrable in arbitrary dimensions if and only if .
As all know, the bifurcation structures of the general integrable systems and autonomous differential systems [11–14] are clear due to the properties of Hamiltonian systems [15–18]. However, the structure for the unintegrable system is not so clear. If takes other forms, what would be the solution of (7)? The regularity of the unintegrable case has been somewhat clarified [19, 20] in regard of the singularity behavior of the equation. Two different finite time blowup solutions of (7) were constructed in , one being an explicit form and the other being an implicit solution. Careful estimation of the two spatial dimension implicit solutions indicates that the energy density decay rate is . However, if the spatial dimension is any integer , does a global smooth (or blowup) solution of (7) exist? Does (7) also present similar regularity aspects under specific smooth initial data? The regularity of dimensional ILL remains open.
Many physical phenomena develop singular behavior, for example, boundary layers or blowup solutions. Liu  analyzed the concentration set of the stationary weak solutions to LLE for the ferromagnetic spin chain. Based on the Ginzburg-Landau approximation, Wang  established the existence of a global weak solution for LLE, , with respect to smooth initial boundary data, which is smooth from a closed set with locally finite dimensional parabolic Hausdorff measure. In 2008, Huh  constructed infinite energy explicit blowup solutions for the modified LLE. Ding  constructed an infinite energy blowup solution for LLE on a hyperbolic target.
The finite energy blowup solution was firstly studied by perturbation methods [25, 26], and it was found that the collapse of a symmetric case with large initial energy formed a singularity where the blowup rate could be estimated by scale invariance. To obtain the required resolution for evolving similar solutions, adaptive mesh refinement (AMR) is required [26–28], which dynamically and locally adds numerical resolution. Although Van Den Berg et al.  present a formal analysis and AMR simulation for LLE with the Gilbert term, rigorous proof of blowup for 2-dimensional (2D) LLE remains open. However, Ding et al.  proved that or dimensional LLE with the Gilbert term will lead to a finite time blowup under specific initial boundary conditions. Some regularity and blowup results for LLE were derived considering the Maxwell field [30, 31].
Although the blowup problem is clear for specific settings (initial state or initial boundary conditions), the blowup problem for ILL is unclear. The one-dimensional ILL will not form a singularity as the inhomogeneous term is periodic . However, if the inhomogeneous term is in some other specific format , blowup occurs for the inhomogeneous HMF.
We studied the blowup and energy gap for dimensional aperiodic ILL on target and investigated what happens as tends to infinity when the initial data is smooth and sufficiently large, in particular whether the solution develops distinct behaviors (finite or global time singularity) under the large data. Global smooth (or blowup) theory for ILL was not established, but we discuss some special solutions that form a singularity in finite time and classify these solutions and analyses based on their energy density.
In Section 2, we obtain a blowup solution for the ILL and derive the exact decay rate of the energy density for that solution. In Section 3, we obtain another local blowup solution that contains an energy gap under the inhomogeneous term and classify this solution into four types. In Section 4, we obtain the decay rate and prove the decreasing (or increasing) property of the local blowup solution for one of these types under some specific coefficient settings. Finally, in Section 5, we summarize the paper and present our conclusions.
2. Blowup Solution on a Sphere
2.1. Blowup Solution Derivation
Selecting the appropriate solution to construct the blowup solution is difficult. Group invariance (under or ) can be applied to search for exact solutions of LLE [34–36], but ILL will not admit any group invariance. Following [19, 20, 37], we use cylindrical coordinates, where , , and are functions to be determined.
For simplicity, consider the case where does not contain and denote , , and as , , and , respectively. Then, from (9), (7) will be greatly simplified and maybe is transformed intoandwhich are nonlinear partial differential equations. To solve these equations, let us assume that is a variable separation function:where and are dependent on and , respectively; and is a constant. Since does not contain , and are denoted as and , respectively. Substituting (12) into (10) and (11),andwhere .
The derivation of (20) is
If and are nonzero constants, regardless of the value of , the left sections of (20) and (21) are continuous functions, and (21) is nonzero. Therefore, according to the implicit function theorem, there exists a function that satisfies (20), and we have the following conclusion.
The energy density of (22) will also blow up, as can be seen from the energy density:
2.2. Decay Rate of the Blowup Solution
From the solutions and the energy density of the upper section, it is difficult to see the decay behavior of energy density. The decay characteristics of the solution determine whether solution energy can be finite value across the whole space. Therefore, we investigate the decay rate of the solution using the implicit solution from Section 2.1.
From (26), , so if and ,where, in this case, is a strictly increasing function in the interval .
Substituting () into the left side of (20),
If and , then, from (20),and if ,
Combining and (27),
From (33), as , the decay rate of the energy density is where is function of ; and as , .
Thus, we may obtain the decay rate of the case. If and , then the decay rate of the energy density can be similarly proven. Therefore, where satisfies and from (27), if , we may estimate the solution where is a function of , where if and only if , then , which has the limit
Thus, is a continuous function with the limit at of . Hence, whether the energy is a finite value depends on the convergence of at . From (20), if , where is constant. Thus,and if , then
We can obtain the initial energy density from (41) by setting :
From (40), the initial energy density is a finite value and if and only if , will be in infinite value. Combining this and the decay behavior as , the energy of the finite spatial area can be estimated as where is a constant and if and only if . Furthermore, the total energy of the spatial region is
The gradient morn and energy density are related: so that, combining Theorem 1 and , as , where is function of , such that if and only if , .
3. Multiple Branches of the Blowup Solutions
We investigate the classification of the blowup solution. Since (7) may contain many solutions different from (22), we extend (22) to a general form to find other solutions. In particular, we use (9) as an undetermined solution and assume that (), , and are constant. We set where and Therefore,
If we set then we can obtain a solution of (56) which satisfies the initial condition :
The derivative of the left of (58) is
We set , , and . Thus, (59) is if and only if , and
Regardless of value, (59) and the left section of (58) are continuous functions, and (59) is not zero. According to the existence theorem of implicit functions, for any there is an that satisfies (58).
From the above process, we obtain the following conclusion.
The energy density of the solution from Theorem 3 is
We must solvewhich can be expressed as
If , and
Solving we obtain
Similarly, solving we obtain
If and , then the inequality is and if and , then
Thus, if , , , and then (68) holds, and since ,
If , , , , and then
From Tables 1 and 2, satisfying (58) can be subdivided into the following categories:(1), and (or , and ). is a continuous curve. If , solving (58), and (56) can be expressed as If the right section of (82) is , then . Hence, as . Thus, the tangent of is perpendicular to the axis at . If , then from (56). Hence, the tangent of is parallel to the axis at . Specifically, in the fourth quadrant of the coordinate system, is monotonically increasing and . In the first quadrant, is monotonically decreasing and . For example, if , , and , then (58) is and the evolution of , determined by (83), is shown in Figure 1.(2), and (or , and ). The evolution of in the first and the fourth quadrant are opposite that of case . In the fourth quadrant, is monotonically decreasing, and in the first quadrant, is monotonically increasing. The tangent of is perpendicular to the axis at . If we set the maximum value of in first quadrant will not exceed . Specifically, if , , , and , then (58) is and the evolution of from (85) is shown in Figure 2.(3), and (or , and ). is completely in the fourth quadrant, and is monotonically increasing. The maximum value of will not exceed , and the curve in this quadrant has lower bound . For example, when , , , , and , (58) is and Figure 3 shows the evolution of from (86) with and .(4), and (or , and ). is monotonically decreasing in the fourth quadrant, with upper bound .
Cases and are multi-value, i.e., for each , There are two values of . We call these multi-branch functions. However, there is only a single monotonic function for cases and .
and cannot be expressed as a Hamiltonian system; that is, there is no function such that
Therefore, (90) are noncompatible systems. Solving the first equation of (90),where is a function that is only dependent on ; and solving the second equation of (90),where is a function that is only dependent on .
Equations (88)-(89) contain more solutions than (58). The evolution of these equations can also be classified into four types as follows:1.Multibranch elliptic case I: , , and (or , , and ).2.Multibranch hyperbolic case I: , , and (or , , and ).3.Multibranch hyperbolic case II: , , and (or , , and ).4.Multibranch elliptic case II: , , and (or , , and ).
To see the evolution of these four solutions directly, we set and , , and as follows:1..2. and .3., , and .4. and .