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Nonlocal Modified KdV Equations and Their Soliton Solutions

Metin Gürses, Aslı Pekcan

arXiv:1711.01588v2nlin.SI

TL;DR

The paper addresses obtaining soliton solutions for nonlocal mKdV equations of all types. It applies Hirota’s method to the coupled mKdV system and then uses Ablowitz–Musslimani reductions. It reports two types of one-soliton solutions and first-type two- and three-soliton solutions.

  • Problem

    The paper addresses how to obtain soliton solutions for all types of nonlocal mKdV equations.

    Source: §1 · p. 6

  • Method

    The authors use Hirota’s direct method for the coupled mKdV system, then apply Ablowitz–Musslimani reductions to obtain nonlocal mKdV solutions.

    Source: §1 · pp. 4, 6

  • Results

    The reduced mKdV system has two types of one-soliton solutions, while the paper gives two- and three-soliton solutions of the first type.

    Source: §1 · p. 6

  • Takeaways & Limitations

    The reduced equations are integrable by construction and have Lax pairs and recursion operators.

    Source: §5 · p. 42

    Two- and three-soliton solutions are presented only for Type 1.

    Source: §1 · p. 6, §3.1.2 · p. 15

Abstract

from arXiv · show

We study the nonlocal modified Korteweg-de Vries (mKdV) equations obtained from AKNS scheme by Ablowitz-Musslimani type nonlocal reductions. We first find soliton solutions of the coupled mKdV system by using the Hirota direct method. Then by using the Ablowitz-Musslimani reduction formulas, we find one-, two-, and three-soliton solutions of local and nonlocal complex mKdV and mKdV equations. The soliton solutions of these equations are of two types. We give one-soliton solutions of both types and present only first type of two- and three-soliton solutions. We illustrate our soliton solutions by plotting their graphs for particular values of the parameters.

1 Introduction

The paper derives a coupled mKdV system from the AKNS formalism and applies local and Ablowitz–Musslimani nonlocal reductions. These reductions yield local mKdV equations and several reflection-symmetric nonlocal equations.

1 Introduction

  • A cubic-polynomial Lax pair in the AKNS formalism yields the coupled modified Korteweg-de Vries system.Its variables q and r are generally complex dynamical variables, and a is a complex constant.Source: §1 · p. 4
  • The first local reduction gives a complex mKdV equation with the condition ā = a, while the second gives the usual mKdV equation without a condition on a.Source: §1 · p. 4
  • Ablowitz–Musslimani reductions produce T-, S-, and ST-symmetric nonlocal complex mKdV equations for the three nonlocal choices of (ε1, ε2).Together with the two signs of k, these give six nonlocal integrable cmKdV equations; the parameter condition on a depends on the reduction.Source: §1 · p. 4
  • A second nonlocal reduction yields only the ST-symmetric nonlocal mKdV equation, for (ε1, ε2) = (−1, −1), without an additional condition on a.Source: §1 · p. 5

1. Real ST-symmetric MKdV equation

The paper uses Hirota solutions of the coupled mKdV system and Ablowitz–Musslimani reductions to obtain soliton solutions for nonlocal mKdV equations. It presents one-soliton solutions of two types, but only first-type two- and three-soliton solutions.

1. Real ST-symmetric MKdV equation

  • The nonlocal cmKdV and mKdV equations are described as integrable, with focusing cases for k < 0 and defocusing cases for k > 0.Source: §1 · p. 5
  • Prior work reported varied exact solutions for nonlocal complex and real ST-symmetric mKdV equations, and some solutions in this paper coincide with those in earlier studies.The cited work includes dark-, W-, and M-type solitons, periodic solutions, and several real mKdV solution types.Source: §1 · p. 6
  • The paper obtains nonlocal complex mKdV and nonlocal mKdV soliton solutions by applying Ablowitz–Musslimani reductions to Hirota solutions of the coupled system.The complex equations include T-, S-, and ST-symmetric cases.Source: §1 · p. 6
  • The paper presents one-soliton solutions of both types, but gives two- and three-soliton solutions only for Type 1.Source: §1 · p. 6

2 Hirota Method for the Coupled MKdV System

The section derives a Hirota bilinear form for the coupled mKdV system and constructs its one-, two-, and three-soliton solution pairs.

2 Hirota Method for the Coupled MKdV System

  • The coupled mKdV equations are rewritten in Hirota bilinear form before the soliton expansions are developed.The passage identifies the bilinear form but does not provide its complete equations.Source: §2 · p. 7

2.1 One-Soliton Solution of the MKdV System

  • The one-soliton construction yields a solution pair with arbitrary parameters k1, k2, δ1, and δ2.The expansions use exponential functions F1 and G1 and set ε = 1.Source: §2.1 · pp. 7–8

2.2 Two-Soliton Solution of the MKdV System

  • The two-soliton construction gives a solution pair whose expressions include four pairwise phase terms and a term weighted by M.The parameters ki, ℓi, δi, and αi for i = 1, 2 are arbitrary complex numbers.Source: §2.2 · p. 9

2.3 Three-Soliton Solution of the MKdV System

  • The three-soliton construction gives F5 and G5 as sums of terms weighted by Vij and Wij, respectively.The coefficients Sij and Qij used in these expressions are given in the Appendix.Source: §2.3 · p. 11
  • The resulting three-soliton solution pair includes single-phase, pairwise interaction, and six-phase terms.The coefficient H of the six-phase term is represented in the Appendix.Source: §2.3 · p. 12

3 Local Reductions of the MKdV System

The coupled mKdV system has two local reductions: one yields the local complex mKdV equation, while the other gives the usual mKdV equation. For the local cmKdV equation, the paper derives two one-soliton types; the Type 2 example is a singular periodic breather.

3 Local Reductions of the MKdV System

  • The two local reductions yield the local complex mKdV equation under a condition and the usual mKdV equation without a condition on a.Source: §3 · p. 12

3.1.1 Type 1.

  • Type 1 one-soliton parameters satisfy the conjugation constraints, leaving four arbitrary parameters in the solution.The stated constraints are ā = a, k2 = conjugate(k1), and e^δ2 = k e^(conjugate(δ1)).Source: §3.1.1 · p. 13

3.1.2 Type 2.

  • Type 2 also has four free parameters despite requiring more conditions, and its one-soliton solution is singular for any parameter choice.The example is described as a singular periodic breather-type wave; the paper uses only Type 1 for two- and three-soliton solutions.Source: §3.1.2 · pp. 14–15

3.2 Two-Soliton Solution for the Local CMKdV Equation:

Applying local reductions to the coupled-system solutions gives two- and three-soliton solutions for local cmKdV, and one-, two-, and three-soliton solutions for local mKdV. For local mKdV, only Type 1 is used for the one-soliton solution because Type 2 gives a trivial solution.

3.2 Two-Soliton Solution for the Local CMKdV Equation:

  • The two-soliton mKdV-system solution is reduced to local cmKdV using parameter constraints, and a particular two-soliton solution is presented and plotted.Source: §3.2 · pp. 15–16, Figure 3 · p. 16

3.3 Three-Soliton Solution for the Local CMKdV Equation:

  • The three-soliton mKdV-system solution is reduced to local cmKdV under parameter constraints, with a particular three-soliton solution presented and plotted.Source: §3.3 · p. 16

3.4 One-Soliton Solution for the Local MKdV Equation:

  • For local mKdV, Type 2 yields a trivial one-soliton solution, so the paper presents Type 1, which is non-singular when |γ1| ≤ 1.A particular one-soliton example is also plotted.Source: §3.4 · pp. 17–18, Figure 5 · p. 18

3.5 Two-Soliton Solution for the Local MKdV Equation:

  • The two-soliton system solution gives a local mKdV solution when ki = ℓi and e^αi = k e^δi for i = 1, 2; a particular solution is plotted.Source: §3.5 · p. 18, Figure 6 · p. 19

3.6 Three-Soliton Solution for the Local MKdV Equation:

  • The three-soliton system solution is reduced to local mKdV under parameter conditions, and the paper presents a particular three-soliton solution and its graph.Source: §3.6 · p. 19

4 Nonlocal Reductions of the MKdV System

The paper applies Ablowitz–Musslimani reductions to obtain soliton solutions of several nonlocal mKdV equations. It treats three nonlocal complex mKdV equations and the real nonlocal mKdV equation.

4 Nonlocal Reductions of the MKdV System

  • The Ablowitz–Musslimani reduction (1.7) is used to obtain soliton solutions for three nonlocal cmKdV equations, with a stated condition.Source: §4 · p. 20
  • The nonlocal solution constructions in this section use distinct reductions for complex and real mKdV equations.Source: §4 · p. 20
  • The reduction (1.12) is used to obtain one- and two-soliton solutions of the real nonlocal mKdV equation.Source: §4 · p. 20

4.1 One-Soliton Solution for the Nonlocal CMKdV Equations:

This section presents Type 1 and Type 2 one-soliton solutions for nonlocal complex mKdV reductions, including symmetry-dependent conditions for singularity or nonsingularity and parameter choices for a Type 2 solution of equation (4.41).

4.1.1 Type 1.

  • Type 1 parameters must satisfy ā = ε1ε2a, k2 = ε2k̄1, and e^δ2 = k e^δ̄1.These conditions ensure the one-soliton solution satisfies the nonlocal reduction.Source: §4.1.1 · p. 21

4.1.1.1 Case a. (T-symmetric): (ε1, ε2) = (−1, 1), r = k¯q(−t, x)

  • For the T-symmetric Type 1 case, solutions can be singular for both focusing and defocusing equations, while the focusing solution is nonsingular when α^3 − 3αβ^2 = 0.The stated singularity condition is α^3 − 3αβ^2 ≠ 0 together with the specified discrete times; a separate example has singularities at (t, x) = (1280(2n + 1)π, −6 ln 2).Source: §4.1.1.1 · p. 22

4.1.1.2 Case b. (S-symmetric): (ε1, ε2) = (1, −1), r = k¯q(t, −x)

  • For the S-symmetric Type 1 case, the solutions can be singular for both focusing and defocusing equations, with examples singular at specified parameter-dependent points.The examples report singularities at (t, x) = (−192 ln 2, 2(2n + 1)π) and (−128 …, 4nπ).Source: §4.1.1.2 · pp. 23–24

4.1.1.3 Case c. (ST-symmetric): (ε1, ε2) = (−1, −1), r = k¯q(−t, −x)

  • For the ST-symmetric Type 1 case, solutions are nonsingular for both focusing and defocusing equations unless the stated singularity condition holds; the reduced solution is nonsingular except at μ = ±2.An example gives a nonsingular solution |q(t, x)|^2 = 2/65 + 16 cos(1/2 x + 1/256 t).Source: §4.1.1.3 · p. 25

4.1.2 Type 2.

  • Type 2 adds parameter constraints to those of Type 1, including k1 = −k̄1ε2, k2 = −k̄2ε2, and two exponential relations involving A and k.The conditions are Ake^(δ1+δ̄1) = 1 and Ae^(δ2+δ̄2) = k.Source: §4.1.2 · p. 26
  • For Type 2 T-symmetric solutions, the derived solution is singular at t = 0 and θ = (2n + 1)π, and nonsingular for t ≠ 0.Here θ = (β + γ)x + ω0.Source: §4.1.2.1 · p. 27
  • For Type 2 S-symmetric solutions, singularities occur at x = 0, θ = 2nπ, while the solution is nonsingular for x ≠ 0.A separate example has singularities at (t, x) = (4(2n + 1)π, 0).Source: §4.1.2.2 · p. 28
  • For Type 2 ST-symmetric solutions, the solution is nonsingular when γ < 1, and the authors also give a particular nonsingular example.The general singularity condition is that f(θ) = e^(2θ) − 2γe^θ + 1 vanishes.Source: §4.1.2.3 · pp. 28–29
  • The Type 2 one-soliton for (4.41) is shown with k1 = 1 2, k2 = 1 4, eδ1 = a1 + ib1 = −3, eδ2 = a2 + ib2 = 3 4, k = −1, and a = 2.Source: Figure 16 · p. 29, §4.1.2.3 · p. 29

4.2 Two-Soliton Solution for the Nonlocal CMKdV Equations:

The section imposes parameter constraints so two-soliton solutions satisfy each nonlocal reduction, then illustrates examples for T-, S-, and ST-symmetric cmKdV equations. The examples include singular waves and, for the ST-symmetric case, a non-singular periodic breather-type wave.

4.2 Two-Soliton Solution for the Nonlocal CMKdV Equations:

  • The two-soliton parameters are constrained to satisfy the selected nonlocal reduction, and the examples plot |q(t, x)|² because its expressions are very long.The reductions considered are T-, S-, and ST-symmetric.Source: §4.2 · p. 30

4.2.1 Case a. (T-symmetric)

  • The T-symmetric examples shown in Figures 17 and 18 are singular two-soliton waves.The examples use parameter choices satisfying the T-symmetric reduction.Source: §4.2.1 · pp. 30–31, Figure 17 · p. 31

4.2.2 Case b. (S-symmetric)

  • The S-symmetric examples shown in Figures 19 and 20 are singular two-soliton waves.The examples use parameter choices satisfying the S-symmetric reduction.Source: §4.2.2 · pp. 31–32, Figure 19 · p. 33

4.3 Three-Soliton Solution for the Nonlocal CMKdV Equations:

The section finds parameter conditions for three-soliton solutions to satisfy the nonlocal reductions, then presents particular examples for the T-, S-, and ST-symmetric cases. The illustrated examples are singular waves, including a singular periodical wave for the S-symmetric case.

4.3 Three-Soliton Solution for the Nonlocal CMKdV Equations:

  • The authors derive conditions for three-soliton solutions to satisfy the nonlocal reduction and present examples with particular parameter values.The examples cover T-, S-, and ST-symmetric reductions.Source: §4.3 · p. 34

4.3.1 Case a. (T-Symmetric)

  • The T-symmetric example is shown as a singular three-soliton wave in Figure 23.The example uses parameters chosen to meet the T-symmetric constraints.Source: Figure 23 · p. 35, §4.3.1 · p. 35

4.3.2 Case b. (S-symmetric)

4.3.3 Case c. (ST-symmetric)

4.4 One-Soliton Solution for the Nonlocal MKdV Equations:

For the real ST-symmetric nonlocal mKdV reduction, the Type 1 one-soliton ansatz gives only the zero solution, so the section uses Type 2 and illustrates several resulting wave forms.

4.4 One-Soliton Solution for the Nonlocal MKdV Equations

  • For r = kq(−t, −x), the Type 1 one-soliton ansatz forces k1 = −k2 and yields only q(t, x) = 0, motivating the Type 2 construction.This reduction gives the real nonlocal ST-symmetric mKdV equation.Source: §4.4 · pp. 37–38
  • With the stated parameter mappings, solution (4.62) can be transformed to earlier one-soliton solutions,, and [7].The paper also notes that obtained rogue-wave and rational solutions of equation (1.14).Source: §4.4 · p. 38
  • The examples include a nonsingular soliton-like solution, a complexiton, and a kink-type solution.The section also plots examples of these wave forms for particular parameter choices.Source: §4.4 · p. 39, Figure 27 · p. 39, Figure 28 · p. 40

4.5 Two-Soliton Solution for the Nonlocal MKdV Equations:

The section constructs two-soliton solutions for the real nonlocal ST-symmetric mKdV equation and illustrates nonsingular, soliton-kink, and breather-type examples; it also states that the same approach yields a three-soliton solution.

4.5 Two-Soliton Solution for the Nonlocal MKdV Equations

  • The reduction constraints are used to obtain two-soliton solutions of equation (1.14).The section then gives examples with different parameter choices.Source: §4.5 · p. 40
  • The plotted Example 30 is a nonsingular two-soliton solution.Figure 30 shows this solution for the stated parameter set.Source: §4.5 · p. 41, Figure 30 · p. 41
  • Examples 31 and 32 represent interactions of a soliton and a kink-type wave.The corresponding solutions are plotted in Figures 31 and 32.Source: §4.5 · p. 41, Figure 31 · p. 41, Figure 32 · p. 42
  • Example 33 is a breather-type wave solution, with the graph shown for |q(t, x)|2.The example uses the parameters specified in the section.Source: §4.5 · pp. 41–42, Figure 33 · p. 42
  • The authors state that the same approach also yields a three-soliton solution of equation (1.14).Source: §4.5 · p. 42

5 Concluding Remarks

The paper studies local and nonlocal reductions of the mKdV system and focuses on soliton solutions. It identifies a conjectured correspondence between local and nonlocal reductions based on its observations of NLS and mKdV systems.

5 Concluding Remarks

  • The authors report two local and two nonlocal reductions, producing T-, S-, and ST-symmetric nonlocal cmKdV equations and a real ST-symmetric nonlocal mKdV equation.They state that the reduced equations are integrable by construction, with Lax pairs and recursion operators.Source: §5 · p. 42
  • The paper presents one-soliton solutions of both identified solution types, but only one type of two- and three-soliton solutions.The authors also plot solutions for particular parameter values.Source: §5 · p. 42
  • From the NLS and mKdV systems, the authors conjecture that a system admitting a local reduction has at least one corresponding nonlocal reduction.They cite reductions r(t, x) = kq̄(t, x) and r(t, x) = kq(ε1t, ε2x) as examples in both systems.Source: §5 · p. 43

6 Appendix

This appendix provides explicit algebraic expressions for coefficients introduced in Section 2.3, including formula fragments in indexed k and ℓ parameters. The supplied passages contain no definitions or explanatory text.

6 Appendix

  • The appendix lists explicit parameter-dependent expressions for the coefficients introduced in Section 2.3.The displayed expressions involve k and ℓ parameters, including indices 1, 2, and 3.Source: §6 · pp. 44, 47
  • The formula fragment includes signed cross-terms involving k1 and k3 with ℓ2 and ℓ3, specifically 2k1k3ℓ2 and -2k1k3ℓ3, supporting the reported signed-product pattern.Source: §6 · p. 48
  • A later sequence likewise gives algebraic fragments in k2, k3 and indexed ℓ parameters, with signs and product terms varying across expressions.The fragments include terms involving k2k3 and multiple ℓ parameters.Source: §6 · p. 49
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