LOCAL AND NON LOCAL SOLUTIONS OF COMPLEX FUNCTIONS.
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https://arxiv.org/pdf/2404.14150
சிக்கல் இயங்கி களில்துண்டு பட்ட ஆனால் சரியான சமன்பாடுகளின் தீர்வுகள் என்பது ஒரு குறிப்பிட்ட இடத்து (லோக்கல்) அமைவுகளில் ஏற்படுதல் தான். ஆனால் சரியற்ற அந்த இயங்கிகளின் தீர்வுகள் "அந்த இடத்து அமைவு"அற்றவை களில் தான் (நான் லோக்கல்) என்பது இங்கு அறியப்பட வேண்டும். இப்படி தீர்வு "இறக்கப்படுத்தல்" (ரிடக்ஷன் ) அதுவும் சிக்கல் இயங்கிகளில் "மாறுதல் செய்யப்பட்ட"(மாடிஃபைடு) வைகளாக இருக்கின்றன.
இவை "சொலிட்டான் தீர்வுகளாக" உள்ளன.மேலும் அவை "ஜோர்டான் குழு "
( ஜோர்டான் பிளாக்ஸ்) எனும் டிடர்மினன்ட் தீர்வுகளாக இருக்கின்றன.அப்படியொரு இயக்ககரமான (டைனாமிகல் ) 1- சோலிட்டான் தீர்வு பற்றி இப்போது பார்க்கலாம்.இதிலேயே "தொடர்விய எல்லை" (கண்டினுவம் லிமிட்ஸ்) களையும் அறியலாம். மேலே குறிப்பிட்ட
" இட அமைவு அற்ற "துண்டுபட்ட அளவுகளின் தீர்வுகளை அறியலாம். இத்தகைய தீர்வுகள் தொகுவியப்பாடுகளில் (இன்டகரேஷன்ஸ்) சிக்கல் இயங்கியங்களில் ஆய்வு செய்யப்படுவதே (நான் லோக்கல் நான் லீனியர்) தேவை ஆகிறது.
Abstract. Cauchy matrix approach for the discrete Ablowitz-Kaup-New(டைனாமிகல் ell-Segur equations is reconsidered, where two ‘proper’ discrete Ablowitz-Kaup-Newell-Segur equations and two ‘unproper’ discrete Ablowitz-Kaup-Newell-Segur equations are derived. The ‘proper’ equations admit local reduction, while the ‘unproper’ equations admit nonlocal reduction. By imposing the local and nonlocal complex reductions on the obtained discrete Ablowitz-Kaup-Newell Segur equations, two local and nonlocal discrete complex modified Korteweg-de Vries equations are constructed. For the obtained local and nonlocal discrete complex modified Korteweg de Vries equations, soliton solutions and Jordan-block solutions are presented by solving the determining equation set. The dynamical behaviors of 1-soliton solution are analyzed and illustrated. Continuum limits of the resulting local and nonlocal discrete complex modified Korteweg-de Vries equations are discussed.
இப்படிப்பட்ட சிக்கல் தீர்வுகளில் "ஸ்கோடிங்கர்" அலைச்சமன் பாட்டை என்னவென்று பார்ப்போம். "அப்ளோவிட்ஸ்",முஸ்ஸ்லிமணி" என்ற இரு ஆய்வாளர்கள் எழுதுவதை அறிவோம். "பேரிடி ..சார்ஜ் ..டைம் சிம்மெட்ரி "
பற்றித்தான் ஆழ்ந்த ஆய்வுகளை எழுதுகிறார்கள்.
1. Introduction In recent decade, the study of nonlocal nonlinear integrable equations lies at the forefront of research in mathematical physics. This is the case since they have been recognized as basic models for describing parity-charge-time symmetry in quantum chromodynamics [1], electric circuits [2], optics [3,4], Bose-Einstein condensates [5], Alice-Bob events [6,7], and so forth. The f irst such equation, is an integrable nonlocal nonlinear Schr¨odinger equation iζt + ζxx +2δζ2ζ∗(−x,t) = 0, δ = ±1, (1.1) proposed by Ablowitz and Musslimani [8] as a special reduction of the Ablowitz-Kaup-Newell Segur (AKNS) hierarchy [9], where and whereafter i is the imaginary unit, and the asterisk ∗ represents complex conjugation. The equation (1.1) has parity-time symmetry owing to the self induced potential V (x,t) = ζ(x,t)ζ∗(−x,t) = V ∗(−x,t), and the corresponding solution stated at distant locations x and −x are directly coupled, reminiscent of quantum entanglement between pairs of particles. From a mathematical point of view, studies of the nonlocal equations are also interesting because these equations often feature in distinctive types of solution behaviors, such as finite-time solution blowup [8,10], the simultaneous existence of solitons and kinks [11], the simultaneous existence of bright/dark solitons [8,12], and distinctive multisoliton patterns [13]. Following the introduction of this nonlocal nonlinear Schr¨odinger equation, many other non local integrable systems have been proposed, one of which is the nonlocal complex modified Korteweg-de Vries (cmKdV) equation [14] Ut +Uxxx +24δUU∗(−x,−t)Ux = 0, δ = ±1, (1.2) Key words and phrases. discrete cmKdV equations, Cauchy matrix reduction approach, solutions, dynamical behaviors, continuum limits. 1 where the nonlocality is of reverse-space-time type. In the past few years, the nonlocal cmKdV type equations started to attract a lot of attention since the local cmKdV type equations them selves have a wide range of physical applications in the propagation transverse-magnetic waves in nematic optical fibers [15] and few-cycle optical pulses [16]. Many methods have been developed to construct a variety of exact solutions for the nonlocal cmKdV type equations. In [17], Yang and Yang introduced a simple variable transformation to construct multisoliton and quasiperi odic solutions for the nonlocal cmKdV equation from those of local cmKdV equation. With the help of the Darboux transformations, various solutions with physical significance of the nonlocal cmKdV equation were obtained in [18], such as dark solitons, W-type solitons, M-type soli tons, and periodic solutions. Based on the double Wronskian solutions for the AKNS hierarchy, bilinearization reduction scheme was developed by imposing a constraint on the two basic vec tors in double Wronskians such that two potential functions in the AKNS hierarchy obey some nonlocal relations, which allowed us to obtain solutions of the nonlocal cmKdV equation [19]. Inverse scattering transform was developed to deal with the multisoliton solutions for a nonlocal cmKdV equation with nonzero boundary conditions at infinity and constant phase shift [20]. Then Riemann-Hilbert technique was applied to study soliton solutions for a generalized non local cmKdV equation, which includes the equation (1.2) as a special case [21]. Hirota method and an improved Hirota bilinear method were also proposed to discuss soliton solutions for the nonlocal cmKdV equation [22,23]. Almost simultaneously, there has been rapid development of research on nonlocal integrable semi-discrete nonlinear systems described by differential-difference equations, involving exact solutions and dynamical behaviors [24–28], gauge equivalence [29, 30], as well as integrabil ity [31,32]. These studies have tremendously increased our understanding about those particu larly rich systems. In recent years, the study of integrable discrete nonlinear systems governed by difference equations (lattice equations) has made clear that these objects are not only interesting for their own sake but are, in fact, more fundamental than their continuous and semi-discrete counterparts (see [33] and the references therein). Quite expectedly, the properties of the non local integrable continuous/semi-discrete nonlinear equations find themselves reflected in the properties of their discrete analogues. The study of nonlocal integrable discrete nonlinear sys tems originates from the proposal of nonlocal Adler-Bobenko-Suris (ABS) lattice equations [34]. For the whole equations in the two-component ABS lattice list [35], they admit reverse-(n,m) and reverse-n nonlocal reductions, respectively. In a series of works [36–38], we introduce the Cauchy matrix and bilinearization reductions schemes for some nonlocal discrete integrable sys tems. As a result, Cauchy matrix solutions to a discrete sine-Gordon equation and double Casoratian solutions to a discrete sine-Gordon equation and a discrete mKdV equation were constructed. These two methods have some resemblances. For instance, both methods are based on the solutions of before-reduced system. In addition, they involve, first taking appropriate reductions to get the nonlocal integrable systems, and second solving the matrix equation algebraically to derive the exact solutions. Even so, the bilinearization reduction scheme is superior to the Cauchy matrix reduction scheme. This is because, in the Cauchy matrix reduction approach, solutions of the original before-reduction AKNS system should satisfy two Sylvester equations [36]. In the real reduction case, the solvability of these two Sylvester equations usually conflicts with the solvability of the Sylvester equation in the matrix equation set (for more detailed explanations, one can refer to the conclusions in [39]). In spite of the imperfection, Cauchy matrix reduction scheme is still an efficient approach to investigate the solutions of nonlocal complex integrable systems [39,40]. The Cauchy matrix reduction scheme has closed connection with the so-called Cauchy matrix approach. The latter 2 one was firstly proposed by Nijhoff et al. to deal with the multisoliton solutions of the ABS lattice equations [41] and subsequently extended by Zhang and Zhao to obtain more kinds of exact solutions beyond soliton solutions [42]. The (generalized) Cauchy matrix approach is a byproduct of direct linearization [43, 44] and arises from the famous Sylvester equation [45]. In the present paper, we are interested in the solutions of nonlocal cmKdV equations reduced from the (third-order) discrete AKNS equations associated with the Cauchy matrix framework introduced in [46]. Noting that the discrete AKNS equation given in [46] doesn’t admit nonlocal reduction, regardless of real reduction or complex reduction. Thus in this paper we will reconsider the Cauchy matrix procedure of the discrete AKNS equation. We will show that there are four discrete AKNS equations derived from the Cauchy matrix framework, some of which admit local reduction, whereas the left ones admit reverse-(n,m) nonlocal reduction. After imposing the local and nonlocal complex reductions, two local and nonlocal discrete cmKdV equations are obtained. Solutions involving multisoliton solutions and Jordan-block solutions for the resulting discrete cmKdV equations are discussed by solving the determining equation set (DES). Continuum limits will be also discussed. The paper is organized as follows. In Section 2, Cauchy matrix approach to the discrete AKNS equation is reestablished. In Section 3, local and nonlocal complex reductions are imposed on the discrete AKNS equations and two local and nonlocal cmKdV equations are listed. Exact solutions, including multisoliton solutions and Jordan-block solutions, are constructed by solving the DES. In Section 4, continuum limits are introduced to discuss the local and nonlocal semi discrete cmKdV equation and the local and nonlocal continuous cmKdV equation. Section 5 is for conclusions. 2. Cauchy matrix approach for discrete AKNS equation In this section we exhibit the Cauchy matrix framework for the discrete AKNS equation. We f irst introduce a DES, including a Sylvester equation and two shift relations of column vector r. We then introduce master functions S(i,j) and list some properties. As a consequence of combination of shift relations with the help of recurrence relations and symmetric property of S(i,j), four discrete AKNS equations are revealed as closed-form. We use the following convention for elementary shifts in the two directions of the lattice: for a dependent variable f defined on the two-dimensional lattice with discrete coordinates (n,m) ∈ Z2, e.g., f = f(n,m), the operations f → f and f → f denote elementary shifts in the two directions of the lattice, i.e., f = f
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