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In the above equations, the ions are assumed to be cold and on the slow ion time scale, the electrons are assumed to be in local thermodynamic equilibrum. When the electron inertia is neglected, the electrons can be considered to follow a Boltzmann distribution if the propagation vector has a small component along the ... |
[MATH] . This enables us to consider propagation perpendicular to the magnetic field with the wave vector [MATH] The linear propagation of electrostatic ion cyclotron waves propagating perpendicular to the magnetic field is governed by the dispersion relation |
[EQUATION] where [MATH] In order to derive the nonlinear evolution equation governing the propagation of the electrostatic ion cyclotron waves, we assume perturbation of the form [MATH] |
exp[i( [MATH] [MATH] [MATH] [MATH] t)], and adopt the reductive perturbation expansion technique. All the physical quantities are expanded about their equilibrium values as- |
[EQUATION] [EQUATION] [EQUATION] where [MATH] denotes the [MATH] and [MATH] components of ion velocities. We have introduced the following stretched variables with asymmetric scaling on transverse direction as |
[EQUATION] where [MATH] is the group velocity in the [MATH] axis. The scaling used here is different from the scaling involved in the derivation of Davey-Stewartson equation that has a symmetric dependence on all the space variables. The stretching in this case is asymmetric with respect to one of the space variables. ... |
Transforming all independent variables by equation ( ), we expand equations ( ) and carry out a systematic balancing of terms at each order of [MATH] The coefficients appearing at different orders are all given at the appendix. |
At [MATH] order we get [EQUATION] Combining the above expressions leads to the linear dispersion relation for the ion acoustic wave- |
[EQUATION] Similiarly at [MATH] we get, [EQUATION] At [MATH] we obtain, [EQUATION] The group velocity along x axis, [MATH] , can be found out from this order of calculation as |
[EQUATION] At [MATH] [EQUATION] At [MATH] order, an NLS-type equation (space co-ordinate [MATH] replacing the time co-ordinate) is obtained as- |
[EQUATION] The above space-type NLS equation has resulted because in the present work, we have scaled the transverse variable [MATH] in the same way as time is scaled in the derivation of NLS equation. |
At [MATH] order we find [MATH] since [MATH] as [MATH] The other quantities determined are- [EQUATION] [EQUATION] [EQUATION] [EQUATION] |
Detailed mathematical forms of all the coefficients occuring in the above equations are given in the Appendix. Similarly the [MATH] order quantities like [MATH] etc can be determined by the same procedure but the exact expressions cannot be given in view of their extreme cumbersome nature. |
Finally at [MATH] order a two dimensional evolution equation is obtained in the form [EQUATION] where the coefficients [MATH] [MATH] , which are real constants dependent on parameters [MATH] and |
[MATH] , are too cumbersome to be expressed in an explicit form. This is a general two dimensional non-integrable equation of two dependent variables [MATH] and [MATH] . If it is assumed that the term |
[MATH] depends on [MATH] like the other terms as [MATH] [MATH] etc, then the only possible consistent relation between [MATH] and [MATH] would be |
[MATH] . Hence we consider [MATH] where [MATH] is a constant dependent on [MATH] . Now using ( 12 ) in ( 17 we see that the general nonintegrable equation ( 17 ) turns into the form |
[EQUATION] for the choice of the constant [EQUATION] where [MATH] depends on the parameters [MATH] In case of the multidimensional extension of modulated ion acoustic wave by Nishinari Nishinari , general multidimensional coupled equations were obtained which were converted to the integrable DS1 equation for the specif... |
Nishinari ). Similarly, in our case, for the specific choice of [MATH] as given in ( 19 ), the general nonlinear nonintegrable equation becomes completely integrable ( 18 ). As the explicit representation of the coefficients [MATH] and [MATH] are too cumbersome, we will show their behavior graphically. In Figure (1), w... |
[MATH] and [MATH] with [MATH] for [MATH] Now rescaling the variables [MATH] and [MATH] in ( 18 ) we get [EQUATION] and renaming [MATH] as [MATH] it gives |
[EQUATION] which is our new (2+1) dimensional completely integrable evolution equation that has been obtained at a higher perturbation order compared to the NLS equation, hence expected to address weaker effects. Similar equation was derived in the context of water waves MyRogue in order to model oceanic rogue waves. I... |
The study of propagation of modulated ion acoustic waves in the presence of a magnetic field has been extensively done using the NLS equation that restricts the study to one dimension. A multidimensional generalization of the NLS equation for a modulated ion acoustic wave packet propagating in a magnetized plasma leads... |
The various properties, as well as the solutions of this new equation are explored in the foregoing sections of the paper. III Modulation instability |
Instability of a planar wave, appearing due to the interplay between dispersion and nonlinear effect called modulation instability (MI) BF which has been in the continuous focus for many years McLean Ramamonjiarisoa |
For investigating the contributions to the frequency due to the linear dispersive and the nonlinear term in 18 ), we insert the plane wave solution [MATH] with [MATH] |
as the real constant amplitude, [MATH] as frequency and [MATH] as the wave vector . For the plane wave to be an exact solution of ( 18 ), the frequency should be [MATH] |
where [MATH] is the frequency due to linear dispersion and [MATH] is its nonlinear correction, which depends on the amplitude of the wave as well as on the x component of the wave vector. |
Now to explore the onset of MI in the system affecting this plane wave solution, we perturb it by a small parameter function [MATH] . Note that the perturbation is considered in both the space directions. |
The solution [EQUATION] neglecting the higher order terms in [MATH] yields from ( 18 ) a linear equation for [MATH] as [EQUATION] |
For detecting the instability of the perturbation we represent [EQUATION] Inserting this form of perturbation in equation ( 23 ) and arranging the independent terms we get a set of two homogeneous equations for the arbitrary coefficients [MATH] nontrivial solutions of which can exist only when the determinant of the ma... |
[MATH] where [MATH] and [MATH] [MATH] which gives finally [EQUATION] Therefore, under the condition [MATH] which is [MATH] with [MATH] , i.e when [MATH] the modulation frequency [MATH] can acquire an imaginary part [MATH] initiating an exponential growth of perturbation with time [MATH] and hence onsetting the MI. [MAT... |
respectively. For plotting the growth rate and stability region, specific values of [MATH] and [MATH] are chosen from Figure (1) as -1.65, 0.46 and -0.49 respectively when [MATH] Both these figures show clearly that the behavior of MI as well as growth rate has a strong directional preference and range. |
The stability plot is drawn in Fig. 3 in the [MATH] plane with the shaded region showing the domain of MI. A comparison with these obtained conditions of MI of ( 21 with other modulated wave equations may be illuminating. Evolution of a wave satisfying one dimensional NLS equation depends on the product of the coeffici... |
These are the discussions in (1+1) dimension following one space directions. The situation becomes more complicated and intricate if the transverse perturbations are included. For example, the propagation of dust ion acoustic waves with combined effect of bounded cylindrical geometry and transverse perturbations Xue le... |
Our system ( 18 ) is also (2+1) dimensional involving asymmetric dependence on the transverse coordinate. From the condition ( 25 ) we see that the growth rate and instability condition is more complicated involving all longitudinal and transverse components |
[MATH] with a strong directional preference and range [MATH] , compared to the one dimensional case. The conditions also involve the coefficients [MATH] of equation ( 18 which are the functions of the system parameters |
[MATH] and [MATH] . The graphical variation of [MATH] with magnetic field [MATH] for given set of [MATH] is shown in Figure-1 from where we can see that numerical value of both |
[MATH] increases as we increase the magnetic field. Since the product [MATH] is negative, modulation instability can set in when the product [MATH] is negative, and the system remains modulationally stable for positive values of [MATH] These are the interesting features of our exact model having distinct structure than... |
IV Connection with Kadomtsev- Petviashvili (KP) equation It is interesting to note that the new integrable equation ( 21 ) together with the space NLS equation derived in ( 12 ), has a deep connection with another wellknown (2+1) dimensional evolution equation. The equation ( 21 ) along with ( 12 ) is a complex equatio... |
Equation ( 21 ) is our new evolution equation after scaling, while the space NLS equation ( 12 at the same scaling becomes [EQUATION] |
where [MATH] are two real constants dependent on [MATH] as [MATH] [MATH] Now, if ( 21 ) is multiplied by [MATH] and its complex conjugate equation by [MATH] and subtracted from one another, then taking derivative w.r.t [MATH] we get another equation containing quadratic power in [MATH] . Now using ( 26 ) and its comple... |
[EQUATION] where [MATH] Equation ( 27 ) is nothing but the well known KP equation where [MATH] being the square of modulus of the wave [MATH] , the dependent variable of equations ( 21 ) and 26 ). It means that the square of the absolute value of the wave without modulation satisfies another real equation which is also... |
For ion acoustic wave, KP equation has been derived and its stability properties under transverse perturbations have been discussed in many plasma systems like quantum electron ion plasma Mushtaq or in electron-positron-ion plasma with high energy tail electron and positron distribution Shahmansouri and in other fields... |
Soliton solutions As a direct feature of integrable system, our equation, 21 ) must admit higher soliton solutions. In this section we will elaborately discuss its multisoliton solutions, which can be derived by many methods e.g, inverse scattering transform, Hirota method and various dressing methods. The IST method i... |
[EQUATION] where [MATH] and [MATH] are complex and real functions respectively. Using ( 21 ) one derives a pair of bilinear equations: |
[EQUATION] Multisoliton solutions are obtained by finite perturbation expansions as [EQUATION] where [MATH] is formal expansion parameter need not to be small. Collecting like powers of [MATH] , we obtain the following series of equations: |
[EQUATION] [EQUATION] [EQUATION] [EQUATION] and similarly higher order equations. V.1 1-soliton To construct 1-soliton solution for 21 ) we assume the ansatz |
[EQUATION] where [MATH] are complex constants. From equation ( 31 ) therefore one obtains the associated dispersion relation [MATH] |
using which the equation ( 32 ) is solved easily to yield [EQUATION] We can verify using ( 35 ) and ( 36 ), that all higher order terms in [MATH] |
beyond [MATH] and [MATH] trivially vanish. Absorbing [MATH] in arbitrary constant [MATH] we construct from 28 ) using ( 35 ) and ( 36 the 1 soliton solution in the form |
[EQUATION] where [MATH] depends on the parameter [MATH] One can identify the interesting 2d nature of our equation ( 21 ) by making [MATH] , then from eq. ( 36 ) we can see that [MATH] will diverge and no soliton solution can be found. If additionally we use the dispersion relation of the constraint equation |
[EQUATION] which comes from ( 12 ), as [MATH] , the soliton solution ( 37 ) simplifies to yield the conventional form [EQUATION] |
A frozen picture of the modulus of our travelling soliton solution ( 39 ) at time [MATH] is shown in Fig. 4. V.2 2-Soliton For obtaining 2-soliton solution we start with the standard procedure assuming |
[EQUATION] where the parameters involved are complex numbers. Applying similar dispersion relations as earlier we get [MATH] and obtain from ( 32 |
[EQUATION] where all the constant parameters can be worked out explicitly (see Appendix ). Similarly equation ( 33 ) at higher order expansion gives |
[EQUATION] where the relevant parameter details are given in Appendix . Using further equation ( 34 ) one obtains [EQUATION] with the relevant parameters presented in Appendix. For simplifying the expressions,as mentioned earlier, we can use the constraint equation ( 38 ), imposing the relations between |
[MATH] [MATH] and [MATH] [MATH] as [MATH] [MATH] (see Appendix). Here we find again, that the higher order terms in [MATH] beyond [MATH] and |
[MATH] trivially vanish, leaving the exact 2- soliton solution in the form [EQUATION] Here also we can see that for no transverse dependence i.e. [MATH] , all the quantities determined [MATH] |
diverges and two soliton solution cannot be found which indicates the strict 2d nature of the equation. graphical plot of the modulus of this solution in [MATH] -dimensions, frozen at time [MATH] , is shown in Fig. 5, where the 2-soliton as two interacting 1-solitons is clearly seen on a 2D [MATH] -plane. Following the... |
These one and two soliton solutions of ( 21 ) have similarities with the soliton solutions of NLS equation with an additional transverse dependence. Since a purely 1D model cannot account for the observed features of many physical situations, specially in auroral region with higher polar altitudes Franz corresponding 2... |
VI Exact static 2D lump solution: Localized wave structure in (2+1) dimensional systems are very important in terms theoretical and experimental aspects of plasma. Localized rational structure following KP-I equation have been found by Janaki et.al in the propagation of oblique magnetosonic wave in warm collisional pla... |
The static 2D rational lump solution is given by [EQUATION] where c , [MATH] are 2 free parameters. From this we can see that the wave attains the maximum amplitude |
[EQUATION] at the centre x=0, y=0 which can be controlled by c. At large distances( [MATH] [MATH] [MATH] [MATH] the amplitude goes to unity. The steepness of this static wave solution as observed from the front is [MATH] is related to another free parameter [MATH] .The amplitude of the wave falls to its minimum at x=0,... |
[MATH] Hence the density gets localized at the centre x=0,y=0 and the concentration can be controlled by the free parameter [MATH] . This is an interesting feature because in actual physical situation the ion density can change which need to be controlled by the free parameters. This is absent in the (1+1) dimensional ... |
can achieve any amplitude and steepness, relevant to the actual physical condition. Thus unlike the exponentially decaying dromion solution of DS1 equation, our system ( 21 ) provides a rational solution in both space directions having similar structure to the rational solution of KP-I equation. There lies another conn... |
VII Other integrable properties: Lax Pair, Conserved quantities Since our new equation ( 21 ) is a completely integrable system, it possesses the associated integrable properties. One of such properties i.e, existence of higher soliton solutions, have been discussed using Hirota bilinearization method in the previous s... |
For equation ( 21 ), one can find the associated linear system [MATH] with a Lax pair given by [EQUATION] where [EQUATION] with [MATH] Pauli matrices. The flatness condition: [MATH] , of the given Lax Pair gives our equation 21 ) together with the space NLS equation like ( 26 ) for particular values of [MATH] and [MATH... |
Systems with infinite degrees of freedom like 21 ), when integrable, should have infinite set of independent conserved quantities, which can be derived explicitly as |
[EQUATION] and so on. Note the involvement of both the space-variables [MATH] in this series of independent conserved quantities, which also gives another argument in favor of the integrability of the 2D nonlinear equation ( 21 ). Nevertheless, the one dimensional integral of these conserved quantities signifies also a... |
VIII Conclusive remarks In this work a completely integrable, (2+1) dimensional, modulated , nonlinear evolution equation has been derived in the ion acoustic wave of magnetized collisionless plasma system. It has been obtained at a higher perturbation order compared to the NLS equation, hence expected to address weake... |
IX Appendix: i) Coefficients appearing in the derivation of the two dimensional integrable evolution equation 21 ), given in section-II: |
[MATH] [MATH] [MATH] [MATH] [MATH] [MATH] [MATH] [MATH] [MATH] [MATH] [MATH] [MATH] [MATH] [MATH] [MATH] ii) Coefficients appearing in the Hirota bilinearization procedure in solving equation 21 ), given in section-V: |
[EQUATION] [EQUATION] [EQUATION] [EQUATION] [EQUATION] For simplifying the expressions we can impose the relations between [MATH] [MATH] and [MATH] [MATH] as [MATH] [MATH] , which would yield [EQUATION] [EQUATION] [EQUATION] |
# Source: arxiv 1806.04478 # Title: On the $t$-adic Littlewood Conjecture # Sections: all # Downloaded: 2026-03-03T02:40:38.988889+00:00 |
On the [MATH] –adic Littlewood Conjecture Abstract The [MATH] –adic Littlewood Conjecture due to De Mathan and Teulié asserts that for any prime number [MATH] and any real number [MATH] , the equation |
[EQUATION] holds. Here, [MATH] is the usual absolute value of the integer [MATH] [MATH] its [MATH] –adic absolute value and [MATH] denotes the distance from a real number [MATH] to the set of integers. This still open conjecture stands as a variant of the well–known Littlewood Conjecture. In the same way as the latter,... |
It is known that [MATH] –LC fails when the ground field [MATH] is infinite. The present article is concerned with the much more difficult case when this field is finite. More precisely, a fully explicit counterexample is provided to show that [MATH] –LC does not hold in the case that [MATH] is a finite field with chara... |
The proof is computer assisted. It reduces to showing that an infinite matrix encoding Hankel determinants of the Paper–Folding sequence over [MATH] , the so–called Number Wall of this sequence, can be obtained as a two–dimensional automatic tiling satisfying a finite number of suitable local constraints. |
Acknowledgements EN wishes to thank Dong Han Kim for discussing Remark 2.4 and related matters with him, to Nishant Chandgotia for a discussion about Section , and to Maynooth University for his two visits there during the time he was working on this project. EN’s research was supported by EPSRC Programme Grant: EP/J01... |
The authors wish to thank the referees for their careful reading of the paper and for suggestions which helped improve its quality, and for investing time in experimenting with our computer program. |
Le premier auteur tient à dédier ce travail à Christian Potier en reconnaissance de ses constants encouragements et de sa passion pour les mathématiques. |
In honorem Christiani Figuli Introduction Let [MATH] be a real number. Denote by [MATH] its usual absolute value and by [MATH] its distance to the set of integers. The famous Littlewood Conjecture from the 1930’s states that for any two real numbers [MATH] and [MATH] , the following equation holds: |
[EQUATION] where the infimum is taken over all non–zero integers. The best–known result towards this conjecture is due to Einsiedler, Katok and Lindenstrauss |
who established that the set of possible counterexamples has Hausdorff dimension zero. It is, however, not even known whether the pair of quadratic irrationalities [MATH] satisfies \tagform@ 1.1 |
De Mathan and Teulié suggested a variant of the Littlewood Conjecture which has since then been known as the [MATH] –adic Littlewood Conjecture . According to the latter, given a prime number [MATH] and a real number [MATH] |
[EQUATION] Here, [MATH] stands for the [MATH] -adic absolute value of the integer [MATH] . Upon writing [MATH] , where [MATH] denotes the [MATH] –adic valuation of [MATH] and where [MATH] is an integer, \tagform@ 1.2 amounts to the following relation : |
[EQUATION] (Note that it is not required in \tagform@ 1.3 that the integer [MATH] should not be divisible by the prime [MATH] . It is easy to see that this does not affect the claimed equivalence between the two formulations of the problem.) Akin to the Littlewood Conjecture, it is known thanks to the work of Einsiedle... |
that the set of possible exceptions to the [MATH] –adic Littlewood Conjecture has Hausdorff dimension zero. For detailed accounts on the Littlewood and the [MATH] –adic Littlewood Conjectures, see |
and the references therein. Both of these conjectures admit natural counterparts over function fields, which have attracted much attention. In order to state them, some terminology and notation are first introduced. |
Let [MATH] be a field. Denote by [MATH] the ring of polynomials with coefficients in [MATH] , and by [MATH] the field of rational functions over [MATH] . The valuation on [MATH] given by the degree of a polynomial extends to a valuation on [MATH] so as to provide an absolute value given by |
[EQUATION] for any [MATH] The completion of the field of rational functions is then the field of formal Laurent series denoted by [MATH] . Explicitly, an element [MATH] can be uniquely expressed as a power series with at most finitely many non–zero coefficients corresponding to positive powers of [MATH] ; that is, it c... |
[EQUATION] where [MATH] is a sequence in [MATH] such that [MATH] . The degree of the Laurent series [MATH] is then the integer [MATH] and its absolute value the quantity |
[MATH] Furthermore, one defines the fractional part of [MATH] as [EQUATION] that is, as [MATH] minus its polynomial part [MATH] With the above notation, the Littlewood Conjecture over Function Fields (LCFF), due to Davenport and Lewis |
, can be stated in complete analogy with the real case as follows: for any [MATH] and [MATH] in [MATH] , the equation [EQUATION] |
holds. Here, the infimum is taken over all non–zero elements in [MATH] . In the same vein, De Mathan and Teulié enunciated the [MATH] –adic Littlewood Conjecture [MATH] –LC), which is the analogue over function fields of the [MATH] –adic Littlewood Conjecture: for any [MATH] in [MATH] , the equation |
[EQUATION] holds. Note that in this statement the variable [MATH] plays the role of the prime number [MATH] in the real case, which is justified by the fact that it can be viewed as an irreducible element in the ring [MATH] ). |
In the case of LCFF, Davenport and Lewis established that the set of exceptions is never empty when the ground field [MATH] is infinite. Their work was complemented by that of Baker |
and several other authors who provided explicit counterexamples in this case. In the other direction, see for explicit constructions of pairs of power series satisfying LCFF. Similarly, for [MATH] –LC, De Mathan and Teulié |
established that the conjecture fails when [MATH] is infinite. Bugeaud and De Mathan later provided explicit counterexamples in this case (they also gave examples of power series satisfying the conjecture in any characteristic). |
Much less is known when [MATH] is finite. Interesting results were proved by Einsiedler, Lindenstrauss and Mohammadi where the positive characteristic analogue of the measure classification results of Einsiedler, Katok and Lindenstrauss |
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