Group-invariant solutions of the (2+1)-dimensional cubic Schrödinger equation

Özemir, Cihangir; Güngör, F.

doi:10.1088/0305-4470/39/12/008

Cited by 7 publications

(7 citation statements)

References 12 publications

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“…For special values of the parameters this ODE can be reduced to the Painlevé IV, II or I equations as well as to ODEs for elliptic functions and their elementary degenerations. The above reductions cover the majority of the cases considered in [52,23,22,21,14,38,28]. Such reductions might be valuable in studying various intermediate regimes of the HNLS equation such as those related to descriptions of rogue waves [31,45].…”

Section: Resultsmentioning

confidence: 98%

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A Universal Asymptotic Regime in the Hyperbolic Nonlinear Schrödinger Equation

Ablowitz¹,

Ma²,

Rumanov³

2017

SIAM J. Appl. Math.

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The appearance of a fundamental long-time asymptotic regime in the two space one time dimensional hyperbolic nonlinear Schrödinger (HNLS) equation is discussed. Based on analytical and extensive numerical simulations an approximate self-similar solution is found for a wide range of initial conditions -essentially for initial lumps of small to moderate energy. Even relatively large initial amplitudes, which imply strong nonlinear effects, eventually lead to local structures resembling those of the self-similar solution, with appropriate small modifications. These modifications are important in order to properly capture the behavior of the phase of the solution. This solution has aspects that suggest it is a universal attractor emanating from wide ranges of initial data.

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Section: Resultsmentioning

confidence: 98%

“…A number of exact solutions based on symmetry reductions using Lie group invariance methods, have been constructed for both the NLS and HNLS eq. [52,23,22,21,14,38]; this was also recently revisited in application to HNLS eq. in [28].…”

Section: Introductionmentioning

confidence: 99%

A Universal Asymptotic Regime in the Hyperbolic Nonlinear Schrödinger Equation

Ablowitz¹,

Ma²,

Rumanov³

2017

SIAM J. Appl. Math.

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“…By this way one can also obtain Bäcklund transformations relating solutions of the equation. We refer the interested reader to [13,14,15] and [16], among many other works on Schrödinger and other type equations. After this overview of the methods we shall make use of, we can turn back to have a look at the material we obtained in the previous section.…”

Section: Analysis Of the Reduced Equationsmentioning

confidence: 99%

On some canonical classes of cubic–quintic nonlinear Schrödinger equations

Özemir

2017

Journal of Mathematical Analysis and Applications

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In this paper we bring into attention variable coefficient cubic-quintic nonlinear Schrödinger equations which admit Lie symmetry algebras of dimension four. Within this family, we obtain the reductions of canonical equations of nonequivalent classes to ordinary differential equations using tools of Lie theory. Painlevé integrability of these reduced equations is investigated. Exact solutions through truncated Painlevé expansions are achieved in some cases. One of these solutions, a conformal-group invariant one, exhibits blow-up behaviour in finite time in L p , L ∞ norm and in distributional sense. they can admit. We would like to mention main results of this paper. We showed that the symmetry algebra L of Eq. (1.1) (equivalently, of Eq. (1.2)) is at most six-dimensional, that is, 1 ≤ dim L ≤ 6. The following results concern the canonical equation (1.1), therefore they actually stand for the (in look) more general family (1.2). R1. Any CQNLS equation within class (1.1) having a 6-dimensional symmetry algebra is equivalent to the quintic constant-coefficient equation with q = q 1 + iq 2 , g = h = 0.R2. Any CQNLS equation within class (1.1) having a 5-dimensional symmetry algebra is equivalent to the cubic constant-coefficient equation with g = g 1 + ig 2 , q = h = 0.R3. The symmetry algebra of the genuine (g and q not both zero) variable coefficient CQNLS equation can be at most 4-dimensional. There are precisely four inequivalent classes of equations in this case.R4. None of these classes can be transformed to the standard constant-coefficient cubic-quintic equation with g = g 1 + ig 2 , q = q 1 + iq 2 , h = 0.

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“…One of them is Lie gorup analysis method and it is well 56 known that this method plays an important role to analyze, finding exact solutions and constructing conservation laws. Lie symmetries of integer order differential equations have been studied by many scientists [14][15][16][17][18]. But studies about the invariance properties of FDEs are quite new.…”

Section: Introductionmentioning

confidence: 99%

Lie Symmetry Analysis and Conservation Laws of Non Linear Time Fractional WKI Equation

San¹

2017

Celal Bayar Üniversitesi Fen Bilimleri Dergisi

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In this study, we considered with the time fractional WKI equation, Lie group analysis method are applied to frational WKI equation with the Riemann-Liouville derivative. The invariance properties of this equation were found. Besides we present corresponding infinitesimal generators for the WKI equation. And then the symmetry reductions are constructed with the Erdelyi-Kober fractional operator. Furthermore, we calculate Lie point symmetries associated the nonlocal conserved vectors utilizing the new conservation theorem method for two different cases of time fractional.

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Group-invariant solutions of the (2+1)-dimensional cubic Schrödinger equation

Cited by 7 publications

References 12 publications

A Universal Asymptotic Regime in the Hyperbolic Nonlinear Schrödinger Equation

A Universal Asymptotic Regime in the Hyperbolic Nonlinear Schrödinger Equation

On some canonical classes of cubic–quintic nonlinear Schrödinger equations

Lie Symmetry Analysis and Conservation Laws of Non Linear Time Fractional WKI Equation

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